A  TKEATISE 


ON  THE 


DIFFERENTIAL  AND  INTEGRAL 


CALCULUS, 


AND    ON    TH 


CALCULUS  OF  YARIATIONS. 


BY  EDWAED  H.  COURTENAY,  LL.  D. 

LATE     PROFESSOR    OF    MATHEMATICS    IN     THE 
UNIVERSITY    OF    VIRGINIA. 

'^  i  B  H  A  U  V 

U NIVKHSITV  () F 
('AlAFOUSW 

A.  S.  BARNES  &   C0MI^:ANY, 

NEW  YORK,  CHICAGO,  AND  NEW  ORLEANS. 
1876. 


ValnaUe  f  oris  liy  Leafling  Ante 

TN  THE 

HIGHER   MATHEMATICS* 

<  ^  > 

Pfof,  Mathematics  in  i/ie   United  States  3fililaiy  s±cademy,    If  est  2'oini, 
CHURCH'S    ANALYTICAL    GEOMETRY. 

Elements  of  Analytical  Geometry,  preserving  the  true  spirit  of  Analysis,  and  rendering  the 
whole  subject  attractive  and  easily  acquired. 

CHURCH'S    CALCULUS. 

Elements  of  the  Diflferential  and  Integral  Calculus,  with  the  Calculus  of  Variations. 
CHURCHi'S    DESCRIPTIYE    G}-E0M:ETRY. 

Elements  of  Descriptive  Geometry,  with  its  applications  to  Spherical  Projections,  Shades 
id  Shadows,  Perspective  and  Isometric  Projections.    3  vols. ;  Text  and  Plates  respectively. 


and 


I>ate  i^rof.  Matfiematics  in  the  University  of  Virginia. 

COURTENAY'S    CALCULUS. 

A  treatise  on  the  Diflferential  and  Integral  Calculus,  and  on  the  Calculus  of  Variations. 


CH^S.    W.    HAOKLEY,    S.  T.  r>.. 

Late  ^rof.  of  Mathematics  and  Astronomy  in  Columbia  College. 

H^^CKLEY'S    TRIGJ^ONOMETRY. 

A  treatise  on  Trigonometry,  Plane  and  Spherical,  with  its  application  to  Navigation  and 
Surveying,  Nautical  and  Practical  Astronomy  wid  Geodesy,  with  Logarithmic,  Trigonomet- 
rical,  and  Nautical  Tables^ / Jy  po6 

"Prof,  of  JVat.  i&  Bxp.   l^hilos.  in  the  U.   S.  Military  Academy ^   West  2*oini. 

BARTLETT'S    SYNTHETIC    IVEECHIANICS. 

Elements  of  Mechanics,  embracing  Mathematical  formulae  for  observing  and  calculating 
the  action  of  Forces  upon  Bodies— the  source  of  all  physical  phenomena. 

BARTLETT'S    .A^NJ^LYTICAL    ]V1EC  HANICS. 

For  more  advanced  students  than  the  preceding,  the  subjects  being  discussed  Analytically, 
by  the  aid  of  Calculus. 

BARTLETT'S    ACOUSTICS    AISTD    Or»TlCS. 

Treating  Sound  and  Light  as  disturbances  of  the  normal  Equilibrium  of  an  analogous  char 
acter,  and  to  be  considered  under  the  same  general  laws. 

BARTLETT'S    ASTRONOMY. 

Spherical  Astronomy  in  its  relations  to  Celestial  Mechanics,  with  full  applications  to  the 
tiurrent  wants  of  Navigation,  Geography,  and  Chronology. 


r>AVIES    *&    I?ECK, 

department  of  Mathematics,   Columbia  College, 

Mathem:atical  dictionary 

find  Cyclopedia  of  Mathematical  Science,  comprising  Definitions  of  all  the  terms  employed 
in  Mathematics— an  analysis  of  each  branch,  and  of  the  whole  as  forming  a  single  science. 

Late  of  the   United  States  Military  Academy  and  of  Columbia  College. 

A    CO]VI3?LETE    COURSE    IN    TNI ATH:EM: ATICS. 

See  A.  S.  Barnes  «fc  Co.'s  Descriptive  Catalogue. 


Entered,  according  to  Act  of  Congress,  in  the  year  1855,  by 

A.  S.  BARNES  &  CO., 

In  the  Clerk's  Office  of  the  District  Court  of  the  United  States  for  the  Southern  District  of  New  York, 

courtenay's  cal. 


C-lf 


EDWARD  H.  COURTENAY 


In  the  publication  of  the  following  Treatise  on  the 
Differential  and  Integral  Calculus  by  Edward  H.  Courtenay, 
two  Institutions  have  an  equal  interest  —  the  Military 
Academy  where  he  was  graduated  in  the  year  1821,  and 
the  University  of  Virginia,  where  he  died  in  the  Fall  of  1853. 

Mr.  Courtenay  was  born  in  the  City  of  Baltimore,  on  the 
19th  of  November,  1803.  He  entered  the  Military  Academy 
as  a  cadet  in  September,  1818,  and  was  the  youngest 
member  of  the  Class  of  that  year. 

The  Course  of  Study  embraced  a  term  of  four  years.  In 
three  years  Mr.  Courtenay  made  himself  highly  proficient  in 
all  the  branches,  and  was  graduated  at  the  head  of  his  class, 
in  July,  1821. 

In  his  initiatory  examination  he  made  a  strong  impression 
on  the  mind  of  the  examiner,  who  remarked,  when  the 
examination  was  concluded,  that  "  a  boy  from  Baltimore,  of 
spare  frame,  light  complexion  and  light  hair,  would 
certainly  take  the  first  place  in  his  class." 

We  transcribe  the  following  record  from  the  Eegister  of 
the  United  States  Military  Academy. 

"  Edward  H.  Courtenay — Promoted  Bvt.  Second  Lieut., 
Corps  of  Engineers,  July  1,  1821. — Second  Lieut.  July  1, 
1821. — Acting  Asst.  Professor  of  Natural  and  Expeiimental 
Philosophy,  Military  Academy,  from  July  23,  1821,  to  Sept. 


iv  EDWARD  H.   COURTENAY. 

1,  1822  ;  and  Asst.  Professor  of  Engineering,  from  Sept  1, 
1822,  to  Aug.  31,  1824.— Acting  Professor  ol  Natural  and 
Experimental  Philosophy,  Military  Academy,  from  Sept.  1. 
1828,  to  Feb.  16,  1829  ;  and  Professor,  from  Feb.  16,  1829, 
to  Dec.  31,  1834. — Resigned  Lieutenancy  of  Engineers,  Feb. 
16,  1829 ;  and  Professorship  of  Natural  and  Experimental 
Philosophy,  Dec.  31,  1834. — Professor  of  Mathematics, 
University  of  Pennsylvania,  from  1834  to  1836. — Division 
Engineer,  New  York  and  Erie  Raih-oad,  1836-37. — Civil 
Engineer,  in  the  service  of  United  States,  employed  in  the 
construction  of  Fort  Independence,  Boston  Harbor,  from 
1837  to  1841.*— Chief  Engineer  of  Dry  Dock,  Navy  Yard, 
Brooklyn,  N".  Y.,  1841-42.— Professor  of  Mathematics, 
University  of  Virginia,  since  1842. — Author  of  Elementary 
Treatise  on  Mechanics,  ti*anslated  from  the  French  of  M. 
Boucharlat,  with  additions  and  emendations,  designed  to 
adapt  it  to  the  use  of  the  Cadets  of  the  U.  S.  Military 
Academy,"  1833. — Degree  of  A.  M.,  conferred  by  University 
of  Pennsylvania,  1834;  and  of  LL.  D.,  by  Hampden 
Sidney  College,  Ya.,  1816." 

*  Mr.  Courtenay,  while  employed  as  Engineer  in  the  construction  of  the  works 
in  Boston  Harbor,  was  associated  with  that  distinguished  officer,  Colonel  Sylvanus 
Thayer,  of  the  Corps  of  Engineers. 

The  year  before  Mr.  Courtenay  entered  the  Military  Academy,  as  a  Cadet, 
Colonel  Thayer  had  been  appointed  Superintendent.  He  was  then  engaged  in 
laying  the  foundation  of  the  system  of  instruction  and  discipline  which  has 
imparted  so  much  reputation  to  that  institution. 

It  was  among  the  most  agreeable  and  cherished  remerabrancesof  Mr.  Courtenay's 
life  that  he  enjoyed  the  entire  confidence  and  friendship  of  so  interesting  and 
distinguished  a  man. 

The  relation  of  principal  and  pupil,  in  a  public  institution  became  the  basis  of 
a  sincere  and  generous  friendship  ;  and  when  the  news  reached  the  north  thai 
Courtenay  was  dead,  no  eye  was  moistened  by  a  tear  of  warmer  sympathy  than 
that  of  the  Superintendent  who  had  guided  his  yOuth  and  admired  his  life. 


EDWARD   H.    COURTENAY.  T 

The  autlior  of  this  notice  examined  Mr.  Courtenay  when 
he  entered  the  Militarj^  Academy,  was  associated  with 
him  in  the  Academic  Board,  and  knew  hira  intimately 
in  all  the  situations  which  he  subsequently  filled;  and 
yet  feels  quite  incompetent  to  do  justice  to  the  memory 
of  so  perfect  a  man  and  so  dear  a  friend. 

The  painter  who  has  a  faultless  form  to  delineate  or  a 
perfect  landscape  to  transfer  to  the  canvas,  is  embarrassed 
by  the  very  perfection  of  his  subject.  He  has  nothing  to 
put  in  opposition  to  the  beautiful — no  shading  that  can  give 
full  effect  to  the  living  light.  Characters  which  afford 
strong  contrasts  are  easily  drawn — it  is  the  perfect  char- 
acter which  it  is  difficult  to  sketch. 

The  intellectual  faculties  of  Professor  Courtenay  were 
blended  in  such  just  proportions,  that  each  seemed  to  aid 
and  strengthen  all  the  others.  He  examined  the  elements 
of  knowledge  with  a  microscopic  power,  and  no  distinction 
was  so  minute  as  to  elude  the  vigilance  of  his  search.  He 
compared  the  elements  of  knowledge  with  a  logic  so  scruti- 
nizing that  error  found  no  place  in  his  conclusions ; — and 
he  possessed,  in  an  eminent  degree,  that  marked  character- 
istic of  a  great  mind,  the  power  of  a  just  and  profound 
generalization. 

His  mind  was  quick,  clear,  accurate  and  discriminating 
in  its  apprehensions — rapid,  and  certain,  in  its  reasoning 
processes,  and  far-reaching  and  profound  in  its  general 
riews.  It  was  admirably  adapted  both  to  acquire  and 
use  knowledge. 

The  intellectual  faculties,  however,  are  but  the  pedestal 


vi  EDWARD  H.   COURTENAY. 

and  shaft  of  the  column — the  moral  and  social  ^  faculties 
are  its  entablature  or  crowning  glory.  It  \s  these  faculties 
vvhicli  shed  over  the  whole  character  a  soft  and  attractive 
radiance,  exhibiting  in  a  favorable  light  the  majesty  of 
intellect  and  the  divine  attributes  of  truth,  justice  and 
beneficence. 

It  was  the  ardent  desire  and  steady  aim  of  Professor 
Courtenay,  during  his  whole  life,  to  be  governed  by 
these  principles,  and  there  are  few  cases  in  which  the 
ideal  and  the  actual  have  been  brought  more  closely 
together.  Modest  and  unassuming  in  his  manners  even 
to  diffidence,  he  was  bold,  resolute  and  firm  in  asserting 
and  maintaining  the  right.  Liberal  in  his  judgments  of 
others,  he  was  exacting  in  regard  to  himself  He  could 
discriminate,  reason,  and  decide  justly  even  when  his  own 
interests  were  involved  in  the  issue.  His  love  of  truth 
and  justice  was  stronger  than  his  love  of  self  or  of  friends. 

His  intercourse  with  others  was  marked  by  the  gentlest 
courtesies.  He  was  an  attentive  and  eloquent  listener. 
Diff'erences  of  opinion,  appeared  to  excite  regret  rather  than 
provoke  argument,  and  his  habitual  respect  for  the  opinions, 
wishes  and  feelings  of  others,  imparted  an  indescribable 
charm  to  his  manners. 

As  a  professor  he  was  a  model.  He  was  clear,  concise, 
and  luminous  in  his  style  and  methods.  Laborious  in  the 
preparation  of  his  lectures,  even  to  the  minutest  facts,  he 
was  at  all  times  prepared  to  impart  information.  His  manner, 
as  a  teacher,  was  highly  attractive.  He  never  by  look,  act, 
word,  or  emphasis  disparaged  the  efforts  or  undervalued 
the  acquirements  of  his  pupils.     His  pleasant  smile  and  kind 


EDWARD   H.   COURTENAY.  Vll 

voice,  when  be  would  say,  "  Is  that  answer  peTfeMy 
correct  ? "  gave  hope  to  many  minds  struggling  \vith  the 
difficulties  of  science  and  have  left  the  impression  of  affec- 
tionate recollection  on  many  hearts. 

At  the  Military  Academy,  on  the  banks  of  the  Hudson, 
where  Mr.  Courtenay  was  educated,  and  where  lie  first 
labored  to  advance  the  interest  of  instruction  and  science, 
his  name  is  recorded  on  the  list  of  distinguished  graduates, 
and  honorably  enrolled  among  the  most  eminent  Professors 
of  that  Institution.  There  his  labors  and  memory  will  live 
long  together. 

At  the  University  of  Yirginia  he  has  left  a  name  equally 
dear  to  that  distinguished  Faculty  of  which  he  was  an  orna- 
ment and  to  the  many  pupils  whom  he  there  taught.  When 
these,  in  later  years,  shall  revisit  their  Ahna  Mater,  to  revive 
iarly  and  cherished  recollections — to  strengthen  the  bonds  of 
early  friendships  and  renew  their  resolves  to  be  good  and 
great,  they  will  find  that  a  wide  space  has  been  made  vacant. 
They  w^ill  realize  in  sorrow  that  a  favorite  professor  has  been 
transferred  from  the  halls  of  instruction  to  the  grove  of  pines 
which  borders  the  town,  and  which  contains  the  remains  of 
the  revered  dead.  Thither  they  will  go,  in  the  twilight  of 
the  evening,  to  visit  the  grave  of  a  man  of  science — their 
able  teacher  and  faithful  friend.  In  reviewing  his  life  and 
contemplating  his  character,  they  will  exclaim — 

''  Mark  the  perfect  mau  and  behold  the  upright ;  for  the 
end  of  that  man  is  peace." 


FisHKiLL  Landing, 
March  10th,  1855 


55.     j 


NOTICE. 

The  following  work  was  left  by  Professor  Courtenay,  in  manuscript,  in 
a  highly  finished  condition  ;  and  yet,  it  must  be  regretted  that  it  could  not 
receive  the  final  corrections  of  the  author.  A  premature  death,  at  the 
meridian  of  life,  placed  the  work  in  other  hands,  and  any  slight  inaccuracies 
of  language  which  may  now  appear,  would  doubtless  have  been  corrected, 
if  the  sheets  could  have  passed  under  the  eye  of  the  author. 

It  is  a  cause  of  thankfulness,  however,  that  the  work  was  entirely  com- 
pleted by  Professor  Courtenay ;  and  in  its  publication  the  plan,  language, 
and  even  the  punctuation,  have  been  followed  with  a  fidelity  due  to  the 
memory  of  a  friend. 

The  work  will  be  found  more  full  and  extensive  than  any  which  has  yet 
appeared  in  this  country  on  the  same  subject ;  and  the  part  which  relates 
to  the  Calculus  of  Variations  will  be  especially  acceptable  to  the  Amencac 
public. 

It  is  perhaps  not  improper  to  add,  that  the  Publishers  have  generously 
offered  to  publish  the  work  on  very  favorable  terms,  and  that  the  protltji, 
whatever  they  may  be.  will  go  to  the  family  of  the  author. 


CONTENTS. 


THE  DIFFEKENTIAL  CALCULUS.    Part  L 


CHAPTER  I. 

PAOR. 

First  Principles            .... 

. 

18 

CHAPTER  n. 

Differentiation  of  Algebraic  Functions 

, 

. 

23 

Examples      ..... 

. 

.                  . 

28 

CHAPTER  m. 

Transcendental  Functions 

, 

, 

31 

Examples  of  Logarithmic  and  Exponential  Functions 

. 

32 

Trigonometrical  Functions 

. 

. 

35 

Geometrical  Illustration 

. 

, 

38 

Circular  Functions   .... 

, 

.                  , 

39 

Examples           ..... 

. 

* 

42 

CHAPTER  IV. 

Successive  Differentiation 

, 

,                  , 

45 

Examples           ..... 

. 

•                  • 

46 

CHAPTER  V. 

Maclaurin's  Theorem    .... 

, 

,                  , 

49 

Examples           ..... 

. 

* 

60 

A  {)plications             .... 

. 

. 

53 

CHAPTER  VI. 

Taylor's  Theorem  ..... 

, 

,                  , 

60 

Examples      ..... 

. 

. 

63 

Applications       ..... 

. 

, 

64 

To  Differentiate  u=F{p,q)  where  p=fx  and  g=fi. 

c 

. 

68 

To  Differentiate  u=F  {p,q,r,Sf  &c.)  where  p,q,r,s, 

&c.  are 

functions  cf 

the  same  variable        .... 

. 

, 

.  70 

To  Differentiate  u = F(p,x)  where  p  =fx 

.. 

« 

71 

Partial  and  Total  Differential  Co-efficients 

, 

, 

71 

Examples      ..... 

. 

,           , 

72 

Differentiation  of  Implicit  Functions     . 

. 

. 

73 

X  CONTENTS. 

CHAPTER  VII. 

ESTIM/LTION    OF    THE    VaLUES    OF     FUNCTIONS    HAVING   THE    InDETEEMINaTE 

Form         .......  77 

The  Form  ?       .            .            .            .            .            .  .            .77 

Examples                  .           ••            .            •            •             .  .            80 

The  Form -^      .            .            .            .            .            .  .            .84 

00 

The  Form  oo  X  0,    .  .  .  .  .  .  .  85 

The  Form  oo  -   oo         .  .  .  .  .  .  ,85 

The  Forms  0»,  oo»,  1"  .  .  .  ...  .  85 

Examples  ...  ...  86 

CHAPTER  Vni. 
Maxima  and  Minima  Functions  of  a  Single  Variable  .  .  90 

Conditions  necessary  to  render  a  Function  of  a  Single  Variable  a  Max- 
imum or  Minimum      .  .  .  .  .  .  .91 

Maximum  and  Minimum  Values  of  an  Implicit  Function  of  a  Single 
Variable  ........       9{i 

Examples      .  ...  .  .  .  .  .  97 

CHAPTER  IX. 
Functions  of  Two  Independent  Variables  .  .  .  Ill 

To  Differentiate  a  Function  of  Two  Independent  Variables  .  114 

To  Differentiate  a  Function  of  Several  Independent  Variables  .  .114 

To  Differentiate  Successively  a  Function  of  Two  Independent  Variables     115 
Implicit  Functions  of  Two  Independent  Variables  .  .  .  116 

Given  u  =  ^z,  and  z  =  F{x,y)  to  Differentiate  u  without  EUminating  z     117 
Elimination  by  Differentiation    .  .  .  .  .  .117 

To  Determine  whether  any  Proposed  Combination  of  x  and  y  is  a  Func- 
tion of  some  other  Combination    .  .  .  .  .120 

Development  of  Functions  of  Two  Independent  Variables  by  Mac- 
laurin's  Theorem         .......     121 

Lagrange's  Theorem  .  .  .  .  .  .  122 

Examples  .  .  .  .         '    .  .  .  .     125 

CHAPTER  X. 
Maxima  and  Minima   Functions  of  Two  Independent  Variables  .     129 

Conditions  necessary  to  render  a  Function  of  Two  Independent  Varia- 
bles a  Maximum  or  Minimum  .  .  .  .  .130 

Examples     ........  133 

CHAPTER  XI. 
Change  op  the  Independent  Variable     .....     137 

Examples     ........  138 

CHAPTER  XII. 
Failure  op  Taylor's  Theorem      .....  141 

Examples     ........  143 


CONTENTS.  xi 

THE  DIFFERENTIAL  CALCULUS.     Part  II. 

APPLICATION   OF  THE    DIFFERENTIAL  CALCULUS    TO  THE 
THEORY  OF  PLANE  CURVES. 

FAGB. 

CHAPTER  I. 
Tanrents  to  Plane  Curves. — Normals. — Asymptotes   .  .  .     147 

Differential  Equation  of  a  Tangent  to  a  Plane  Curve  .  .  148 

Differential  Equation  of  a  Normal  to  a  Plane  Curve      .  .  .     149 

Expressions  for  the  Tangent  and  Subnormal  .  .  .  150 

Applications  to  the  Parabola,  Ellipse,  ant^  Logarithmic  Curve  .  .     150 

Expressions  for  the  Tangent,  the  Normal,  and  the  Perpendicular  on  the 
Tangent  of  a  Plane  Curve      .  .  .  .  ..IT)! 

Expressions  for  the  Polar  Subtangent,  Subnormal,  Tangent,  Normal, 
and  Perpendicular  to  the  Tangent  of  a  Plane  Curve  referred  to  Polar 
Co-ordinates  .  .  .  .  .  .  .  15'2 

Examples.     The  Spiral  of  Archimedes.     The  Logarithmic  Spiral.     The 
Lemniscata  of  Bernouilli         ......     15-1 

Rectilinear  Asymptotes         .  .  .  .  .  .156 

Conditions  that  Determine  the  Existence  of  Rectilinear  Asymptotes    .     15G 
Applications  to  the  Hyperbola,  the  Logarithmic  Curve,  the  Cissoid,  the 
Parabola,  the  Hyperbolic  Spiral,  the  Spiral  of  Archimedes,  the  Loga- 
rithmic Spiral,  and  the  Lituus       .  .  .  .  .15? 

Circular  Asymptotes      .  .  .  .  .  .  .159 

CHAPTER  n. 

ClRVAlURE    AND    OsCULATION    OF     PlANE    CuRVES  .  .  .  160 

Differential  of  the  Arc  of  a  Plane  Curve  in  Terms  of  the  Differentials 
of  the  Co-Ordinates    .  .  .  .  .  .  .161 

Conditions  of  the  Osculation  of  Cuprv^s,  in  the  Different  Orders  of  Contact     162 
To  Determine  the   Radius  of  Curvature  and  the  Co-Ordinates  of  the 
Centre  of  the  Osculatory  Circle  .  .  .  .  .164 

Examples      ........  167 

At  the  Points  of  Greatest  and  Least  Curvature  of  any  Curve,  the  Oscu- 
latory Circle  has  Contact  of  the  Third  Order  .  .  .     169 
Curves  Intersect  at  the  Point  of  Contact,  when  the  Order  of  Contact 

is  Even  ;  but  do  not  Intersect  when  it  is  Odd       .  .  .  170 

Formula  for  the  Radius  of  Curvature  when  the  Independent  Variable  is 
Changed  ........     171 

Radius  of  Curvature  of  Curves  referred  to  Polar  Co-Ordinates  .  172 

Examples  ......••     173 

Radius  of  Curvature  of  Curves  referred  to  the  Radius  Vector  and  the 
Perpendicular  on  the  Tangent       .  .  .  .  .174 


Xll  CONTENTS. 

PAGE 

.  CHAPTER,  III. 

EVOLUTES    AND    INVOLUTES       .......        176 

To  Determine  the  Evolute  of  a  Given  Curve  y  =  Fx             .             .  176 

Applications  to  the  Parabola,  Ellipse  and  Equilateral  Hyperbola            .  176 

Normals  to  the  Involute  are  Tangents  to  the  Evolute           .             .  178 
The  Difference  of  any  Two  Radii  of  Curvature  is  Equal  to  the  Inter- 
cepted Arc  of  the  Evolute       .             .             .             .             .             .179 

Evolutes  of  Polar  Curves  Given  by  the  Relation  between  the  Radius 

Vector  and  the  Perpendicular,  on  the  Tangent      .             .             .  180 
CHAPTER  IV. 

Consecutive  Lines  and  Curves     ......  182 

To  Determine  the  Points  of  Intersection  of  Consecutive  Lines  or  Curves  182 
Consecutive  Normals  to  any  Plane  Curve     .             .             .             .183 

The  Curve  which  is  the  Locus  of  the  Points  of  Ir^tersection  of  a  Series 
of  Consecutive  Curves,  touches  each  Curve  of  the  Series,  and  is  called 
the  Envelope  .  .  .  .  .  .  .  .181 

Examples  of  the  Determination  of  Envelopes            ,             .            .  185 
CHAPTER  V. 

SiN'JULAR  Points  of  Curves           ......  190 

Multiple  Points        .  .  .  .  .  .  .190 

To  Determine  whether  a  Curve  has  Multiple  Points  of  the  First  Species  190 

Examples            ........  191 

To  Determine  whether  a  Curve   has  Multiple  Points  of  the  Second 

Species     ........  194 

Conditions  for  Determining  Conjugate  or  Isolated  Points  .  .195 

Examples      ........  196 

Cusps     .........  198 

Examples      ........  199 

Points  of  Inflexion  .  .  .  .  .  .  .201 

Examples      ........  202 

Stop  Points         ........  203 

Shooting  Points        .......  203 

Points  of  Contrary  Flexure  of  Spirals    .....  204 

CHAPTER  VI. 

CtRviLiNEAR  Asymptotes           ......  205 

Conditions  necessary  to  Render  Two  Curves  Asymptotes  to  each  other  205 
General  Form  for  the  Value  of  the  Ordinate  in  Curves  that  admit  of  a 

Rectilinear  Asymptote             .             .             .             .             .             ,  206 

Examples      ........  206 

CHAPTER  VII. 

Tracing  of  Curves               .......  208 

General  Directions    ....,.,  208 

Examples            ........  209 


CONTENTS.  Xlil 

THE  DIFFERENTIAL  CALCULUS.     Paet  IIL 

THEORY   OF  CURVED  SURFACES. 

PA6S. 

CHAPTER  I. 

TaSgent  and  Normal  Planes  and  Lines  ....  214 

General  Differential  Equation  of  a  Tangent  Plane  .  .  .        -  214 

Equations  of  a  Line  Normal  to  a  Curved  Surface  .  .  .  216 

Equations  of  a  Line  Tangent  to  a  Curve  of  Double  Curvature         .  217 

Equation  of  a  Plane  Normal  to  a  Curve  of  Double  Curvature    .  .  217 

Examples  of  Tangent  Planes  to  Surfaces     ....  218 
CHAPTEP.  11. 

Cylindrical  Surfaces,  Conical  Surfaces,  and  Surfaces  of  Revolution  219 

General  Differential  Equation  of  all  Cylindrical  Surfaces    .  .  219 

Equation  of  a  Cylindrical  Surface  which  Envelopes  a  Given  Surface, 

and  whose  Axis  is  Parallel  to  a  Given  Line  ....  220 
When  a  Cylinder  Envelopes  a  Surface  of  the  Second  Order,  the  Curve 

of  Contact  is  an  Ellipse,  Parabola,  or  Hyperbola  .  .  222 

General  Differential  Equation  of  Conical  Surfaces       .  .  .  222 

Equation  of  a  Conical  Surface  which  Envelopes  a  Given  Surface,  and 

whose  Vertex  is  at  a  Given  Point  ....  224 

When  a  Cone  Envelopes  a  Surface  of  the  Second  Order,  the  Curve  of 

Contact  is  an  Ellipse,  Parabola,  or  Hyperbola  .  .  .  225 

General  Differential  Equation  of  Surfaces  of  Revolution     .  .  226 

A  Curved  Surface  Revolving  about  a  Fixed  Axis ;  to  Determine  the 

Surface  which  Touches  and  Envelopes  it  in  Every  Position  .  227 

Examples      .  ......  227 

CHAPTER  HL 

Consecutive  Surfaces  and  Envelopes     .....  230 

Equations  of  the  Intersection  of  Consecutive  Surfaces          .  .  231 

The  Locus  of  the  Intersections  of  a  Series  of  Consecutive  Surfaces 

Touches  each  Surface  in  the  Series    .  .  .  .  .231 

Examples     ........  233 

CHAPTER  IV. 

Curvature  of   Surfaces     .......  235 

Conditions  Necessary  for  Contact  of  the  Different  Orders    .  .  235 

The  Ellipsoid,  Hyperboloid  and  Paraboloid  can  have  Contact  of  the 

Second  Order  .......  236 

To  Determine  the  Radius  of  Curvature  of  a  Normal  Section  of  a  Surface  236 
The  Sum  of  the  Curvatures  of  Two  Normal  Sections  through  the  same 

Point,  and  Perpendicular  to  Each  Other,  is  Constant        .  238 

Principal  Radii  of  Curvature  at  a  Given  Point  .  .  .  239 

Properties  of  the  Principal  and  Normal  Sections     .  .  .  249 


XIV  CONTENTS. 

PAOB 

At  Every  Point  of  a  Curved  Suiface  a  Paraboloid  nay  be  Applied, 

having  Contact  of  the  Second  Order  .....  242 

Meusnier's  Theorem  ......  244 

Lines  of  Curvature        .......  244 


THE  INTEGEAL  CALCULUS.    Part  L 

CHAPTER  I. 

FiBST  Principles      ........  248 

Integration  of  Simple  Algebraic  Forms         ....  249 

Examples  ........  250 

CHAPTER  II. 

Elementary  Transcendental  Forms  .....  253 

Logarithmic  Forms         ...,,,.  253 

Examples      ........  253 

Circular  Forms   .             .             .             .             .             .             .             .  254 

Examples      .             .             .             .             .            .             .      '       .  257 

Trigonometrical  Forms  .......  259 

Examples      .             .             .             .             .            ,            •            .  259 

Exponential  Forms         .             .             .             .            .            .             .  260 

Examples        ........  260 

CHAPTER  in. 
Ratiuis.i.  Fractions  .......     261 

Casx  I.  When  the  Denominator  can  be  resolved  into  Real  and  Unequal 
Factors  of  the  First  Degree  .  .  .  .  .261 

Case  II.  'When  the  Denominator  contains  Equal  Factors  of  the  First, 
Degree  .  .  .  .  .  .  .  .265 

Examples      .  .  .  .  .  .  .  .266 

Case  III.  When  the  Simple  Factors  of  the  Denominator  are  Imagin- 
ary    .  .  .  .  .  .  .  .  .268 

General  Examples    .......  271 

CHAPTER  IV. 

Irrational  Fractions  .......  274 

Case  I.  When  the  Fraction  contains  only  Monomial  Terms  .  274 

Case  II.  When  the-  Surds  in  the  Expression  have  no  Quantity  under 

the  Radical  Sign,  but  of  the  Form  (a  +  ix)  .  .  .  275 

Case  III.  When  there  are  no  Surds  except  of  the  Form  (a+bx-h+c^x^)  277 

Examples     ........  277 


CONTENTS. 


CHAPTER  V. 
Binomial  Differentials      ,  .  .  . 

Conditions  of  Integrability   . 
Examples 

CHAPTER  Vr. 

Formula  of  Reduction 

Formula;  (^),  (5),  (C)  and  (Z)) 

Applications  of  Formulae  {A),  (B),  (C)  and  (D) 

CHAPTER  Vn. 

Logarithmic  and  Exponential  Functions 
Logarithmic  Functions 

Examples  ..... 

Exponential  Functions 
Examples  .  .       '      . 

CHAPTER  Vm. 

Trigonometrical  and  Circular  Functions 

Trigonometrical  Functions 

Formulae  (E),  {F),  (G),  {H).  (I)  and  (K)     . 
'         Applications  of  FormuljE  {E),  {F),  (G),  (H),  (J)  and  {K) 

Formulae  (L)  and  (TJf) 

Circular  Functions  .  .  . 


CHAPTER  IX. 


Approximate   Integration 
Method  by  Expansion    . 
Examples 
Bernouilli's  Series 


CHAPTER  X. 

Integration  between  Limits  and  Successive  Integration 
Integration  between  Limits        .... 
Precise  Signification  of  /   Xdx         .  .  .  , 

Successive  Integration  ..... 
To  Develop  the  n'^  Integral  of  j'^Xdx'"'  in  a  Series 
To  Deduce  the  Development  off^Xdxn  from  that  of  X 
Examples        ...... 


XV 

PA6B. 

28C 
281 
281 


284 
286 


291 
291 
291 
295 
296 


298 
298 
290 
305 
309 
310 


312 
312 
315 

316 
316 
317 
319 
319 
320 


XVI  CONTENTS. 

PAOX. 

THE  INTEGRAL  CALCULUS.    Part  IL 

RECTIFICATION  OF  CURVES.   QUADRATURE  OF  AREAS.   CUBA- 
TURE  OF  VOLUMES. 

CHAPTER  I. 

Rectification  of  Curves   .......  322 

Formula  for  the  Length  of  the  Arc  of  a  Plane  Curve  referred  to  Rec- 
tangular Co-ordinates  ......  322 

Examples  of  its  Application  .....  323 

To  Determine  what  Curves  of  the  Parabolic  Class  are  Rectifiable         .  326 

Formula  for  the  Rectification  of  Polar  Curves  .  .  .  327 

Examples  of  its  Application       ......  327 

Formula  when  the  Curve  is  referred  to  the^Radius  Vector  and  the  Per- 
pendicular on  the  Tangent  .....  328 

CHAPTER  n. 

Quadrature  of  Plane  Areas        .            .            .            ,            .  .    330 

Formulae       ........  829 

Examples           .             .             .             .             .             .             .  .330 

Formula  for  Polar  Curves     .             .             .             .             .             .  333 

Examples           .             .             .             .             .             .             .  .334 

Formula  for  Curves  given  by  Relation  between  the  Radius  Vector  and 

the  Perpendicular  on  the  Tangent             ....  335 

Examples           .             .             .             .             .             .             .  .335 

CHAPTER  m. 
Quadrature  of  Curved  Surfaces        .  .  .  -  .  337 

Formula  for  Surfaces  of  Revolution       .....     338 

Examples      .  .  .  .  .  .  .  .338 

Formula  for  any  Curved  Surface  referred  to  Rectangular  Co-ordinates     340 
The  Tri-rectangular  Triangle  and  the  Groin       ....     342 

CHAPTER  IV. 

Curvature  of   Volumes  ......  344 

Volume  Generated  by  the  Revolution  of  a  Plane  Figure  about  an  Axis  344 

Examples  .....  .  .  345 

Volume  of  all  Solids  Symmetrical  with  respect  to  an  Axis  .  347 

Examples  .  .  .  .  .  .  .  .348 

Volume  of  a  Solid  Bounded  by  any  Curved  Surface  referred  to  Rectan- 
gular Co-ordinates  ......  350 

Examples  ........  35^ 

Volume  of  a  Solid  Bounded  by  a  Surface  whose  Equation  is  referred 

to  Polar  Co-ordinates         ......  353 

Examples  ....  354 


CONTEXTS.  XVll 

THE  INTEGRAL  CALCULUS.    Part  III 

[NTEGRATION  OF    FUNCTIONS   OF    TWO   OR    MORE  VARIABLES. 

CHAPTER  I. 

PAGE. 

[ktegration  of    Expressions    Containing    Several    Independent   Va- 
riables          .            .            .            .            .            .            •            •.  3.56 

Conditions  of  Integrability  of  Exact  Differentials    .             .             .  356 

Formula  for  the  Integration  of  the  Form  du  =  Pdx  +  Qdy       .             .  358 

Examples      .             .                          .             •             •             .             .  361 

Homogeneous  Exact  Differentials          .....  363 

Examples      ........  365 

CHAPTER  II. 

Differential  Equations     .             .            .            .            .       '    .            .  367 

Differential  Equations  of  the  1st  Order  and  Degree               .             .  368 

1st  Case  in  which  the  Variables  may  be  Separated,  Ydx  +  Xdy  =  0  .  369 
^D  CxsE,  the  Form  XYdx  +  XiY^dy  ==  0   .            .            .            .369 

3d  Case,  the  Homogeneous  Form  (xny">  +  axn+'iym—\  ....  -\-  pxn+c 

ym-c)dx     ........  370 

4th  Case,  the  Form  (a  +  bx  +  cy)dx  +  {a-i  +  b]X  +  Ciy)dy  =  0           .  371 

I         6th  Case,  the  Linear  Equation,  dy  +  Xydx  =  X^dx           .             .  372 

6th  Case,  Riccati's  Equation,  dy  +  by^dx  =  ax"idx       .             .             .  375 

Factors  Necessary  to  Render  Differential  Equations  Exact              .  382 

•   Geometrical  Applications  of  Differential  Equations  of  the  1st  Order 

and  Degree     .             .             .             .             .             .             .            •.  389 

CHAPTER  in. 

Differential    Equations    of    the   First    Order  and  of    the   Higher 

Degrees    ........  394 

dy 

When  the  Equation  can  be  Resolved  with  Respect  to  -^           .             .  394 

dy 
When  it  Cannot  be  Solved  with  Respect  to  j-,  but  Contains  only  One 

of  the  Variables,  and  may  be  solved  with  respect  to  it           .             .  397 

2d  Case,  when  the  Equation  is  Homogeneous  with  Respect  to  x  andy  399 

3d  Case,  the  Form  y  =  x^  +  0(^^)in  which  ^  (^)  does  not  con- 
tain X  or  y              .             .             •             •             •             •             •  ^^^ 
4th  Case,  the  Form  y  =  Px -\-  Q,  where  P  and  Q  are  functions  of  p^  .  401 

CHAPTER  IV. 

Singular  Solutions  of  Differential  Equations       .             .             .  403 

Conditions  that  Render  a  Singular  Solution  Possible     .  .  .404 

Singular  Solutions  Illustrated  Geometrically             .             .             .  406 

Conditions  for  Finding  Singular  Solutions  without  First  Determining 

the  Complete  Primitives          .             .             .             .             •             •  407 


xviu 


CONTENTS. 


CHAPTER  V. 
Integraiion  of  Differential  Equations  of  the  Second  Ordei 

The  Fonn  f(x,  3^^)  =::  0 

The  Form  F  (y,    ^)  .=  0 

TheF..^-^       g)=0     .  .  .  .  . 

^''eFo™^(.      I     g)=0      .... 

^^^yorr.  F(y.     %     g)  =  0  .  .  .  . 

The  Form  F{v,  2,  p)  =  0     . 

CHAPTER  VI. 

DjPTfiRENTIAL    EQUATIONS    OF    THE    HiGHER    OrDERS 

The  Form  4^-1",     ^)  =  0      .... 

\drn         dr."— I 


The 


dxn       dz" 
fd"y       d"—'-y> 


FormF(^,      j^  =  0 


CHAPTER  VII. 

Sp    utaneous  Differential  Equations 

Examples  of  the  Integration  of  Several  Systems  of  Equations 


411 
411 

412 

413 

413 

415 
416 

418 
418 

418 


420 
420 


CALCULUS  OF  VARIATIONS. 

CHAPTER  I. 

'?       ct,  anp  First  Principles              .....  425 

{leneral  Principles          .......  425 

A-ppIio -uJ 'ons               .......  427 

CHAPTER  II. 

Aii/.icAf./.'- 8  OF  General  Formula  to  Functions  of  One  Variaile  434 

CHAPTER  III. 

Succes.'  "K  Variation    .......  448 

CHAPTER  IV. 

Maxima  »nd  Minima             .......  461 

Mfjt»ma  and  Minima  of  One  Variable           ....  461 

Re'/ative  Maxima  and  Minima  of  One  Variable  .             .             .             >  475 

Applications               .......  46ft 


1^1  BRA  l;  V 

UNI  VKKs  IT\    OK 

rALIlOUXlA 
DIFFERENTIAL   CALCULUS. 


CHAPTER  I. 


FIRST    PRINCIPLES. 


1.  In    all   mathematical   calculations,    the  quantities   which   are 
^presented  for  our  consideration  belong  to  one  of  two  remarkable 

classes  :  namely,  constant  quantities,  which  are  such  as  preserve 
the  same  values  throughout  the  limits  of  one  investigation;  or 
variable  quantities,  which  may  assume  successively  different  values, 
the  number  of  such  values  being  unlimited. 

The  first  letters  of  the  alphabet,  as  a^  b,  c,  dec,  are  usually 
employed  to  denote  constant  quantities,  and  the  last  letters  z,  y,  .r, 
&c.  are  used,  to  represent  such  quantities  as  are  variable. 

2.  When  two  quantities  x  and  y  are  mutually  dependent  upon 
each  other,  so  that  a  knowledge  of  the  value  o'f  one  will  lead 
to  that  of  the  other,  they  are  said  to  be  functions  of  each  other. 
Thus,  in  the  equations 

y  =  ax,       7/  =z  bx"^  -{-  ex  -\-  e,       y  =  ax^  4-  bx"^  —  ex  -j-  e, 

the  value  of  y  is  determined  as  soon  as  that  of  x  is  known ;  and 
accordingly  y  is  said  to  be  a  function  of  x. 

In  like  manner,  an  assumed  value  of  y  will  fix  the  correspond- 
ing values  (if  .T,  and  therefore  a;  is  a  function  of  y.  There  is 
this   difference,  however,  between   the  two  cases :    when  the  value 


14  DIFFERENTIAL  CALCULUS. 

of  X   is   assumed,   that   of  y   is   obtained   by    a   si  nple    substitu- 

tion ;    whereas   the   determination   of   the   value   of   x   from   that 

of  y   requires   the   solution   of  an   equation.     Hence,  y   is   called 

an   explicit  function  of  x^  but   x   is   said   to  be   an   implicit   func- 

tion  of  y. 

The  general  fact  that  y  is  an  explicit  function  of  x   is  written 

thus  : 

y  —  Fx,        or        y  =  9a:, 

when  the  character  F  or  (p  stands  as  the  representative  of  certain 

operations  to  be  performed  on  the  quantity  x,  the  result  of  which 

operations   will  be   a  quantity   equal  in  value  to  y.      And  when 

we  wish  to   imply  that  the  values  of  x  and  y  are  connected  by 

an  unresolved   equation,  or  that   y  is    an    implicit  function   of  x, 

we  write 

F(x,  y)  =  0,         or         (p{x,  y)  =  0. 

For   the   purpose   of  illustration,  let  there   be   taken   the  three 

equations 

yz=ax-\-b         (1), 

y^ax^i-bx-i-c         (2), 

y  =  ax^-{-  bx^  -{- ex  -\-  e         (3), 

and  suppose  x  to  receive  an  increment  h  in  each  equation,  con 
verting  it  into  x  +  ^h  and  causing  y  to  assume  a  new  value  y,. 
Then  if  the  form  of  each  function,  or  value  of  y,  be  supposed 
to  remain  unchanged,  the  three  equations  (1),  (2),  and  (3),  will 
beccme  respectively 

yi  =  a{x-{-h)\-b         a) 

y,  =  a{x  +  hy  +  b{x  +  A)  +  c         (.5), 

and  yi  =  a{x  +  hf  +  b{x  +  hy  -h  c(x  +  h) -j-  e  («). 

Subtracting  (1)  from  (4)  we  obtain 

y^-~  y  -zz.  ah     (7V 


FIRST  PRINCIPLES.  15 

From  (2)  and  (5)  we  get 

yx-y  =  a{2xh  +  h"^)  +  bh    (8). 
And  from  (3)  and  (6) 

V,  -  y  =  a(Sx^h  +  3xh^  +  h^)  -f-  b{2xh  +  h"^)  +  ch     (9). 
From  (7)  we    ieduce,  by  division,     s^ 


h 


(10); 


and  from  (9) 


from  (8)  ^^-j-^  =  a{2x  +  A)  +  ^     (H) ; 

z=z2ax  -^  ah  +  b'y 

^  =  a{Sx^  +  3;rA  +  h^)  +  b{2x  +  k)  +  c         (12). 

The  results,  (10),  (11),  and  (12),  express  tne  ratio  between 
the  increment  h  assigned  to  x,  and  the  corresponding  increment 
y\  —  y  imparted  to  y.  The  values  of  this  ratio,  in  the  three 
examples  selected,  present  remarkable  differences. 

In  ^he  first  example,  this  ratio  retains  the  same  value  a,  what 
ever  may  be  the  value  assigned  to  the  increment  h.  In  the 
second  example  it  consists  of  two  parts, 

one  =  2ax  -\-  6, 

entirely  independent  of  A,  and   the  other  =  aA, 

which  varies  with  h.  If  the  value  of  h  be  supposed  to  diminish, 
the  ratio 

2ax-^b-\-ah         (11), 

will  become  more  and  more  nearly  equal  to  2ax  +  b ;  and,  final- 
ly, when  h  becomes  indefinitely  small,  the  ratio  is  reduced  to 
this  latter  value. 

The   corresponding  increments  h  and  y,  —  y,  when   indefinitely 

/TV         or  THE         1^ 


16  DIFFERENTIAL    CALCULUS. 

small,  are  called   the  differentials  of  the  quantities  x  and  y,  and 
the  limiting  value  of  the  ratio 


h 
.is  called  the  differential  coefficient^  because  it  is  the  multiplier  of 
the  diflcrential  of  x  necessary  to  produce  the  differential  of  y. 

The  differentials  of  x  and  y  are  written  dx  and  rfy,  the  char- 
acter d  being   the   symbol  of  an  operation    to   be   performed   on 

dy 
X  or  y,  not  a  fixctor :  and  the  differential  coefficient  is  written  -— 

ax 

Moreover,  one  of  the  variables  (usually  x)  is  called  the  inde- 
pendent variable,  its  increment  dx  (although  small)  being  arbi 
trary  ;  while  the  other  y,  whose  increment  dy  depends  on  that 
of  x,  is  called  the  deijeiident  variable  or  simply  the  function. 

In  the  third  example,  the  ratio 

Vi  —  y 
h 

reduces,  at  the  limit  when  ^  =  0,  to 

%  =  3«^2  4.  2bx  +  c. 
dx 

These  examples  illustrate '  the  fact  that  two  indefinitely  small 
quantities  may  yet  have  a  finite  ratio  ;  and  they  suffice  to  show 
that  the  form  of  the  differential  coefficient,  which  is  usually  a 
function  of  x,  will  depend  very  materially  on  the  form  of  the 
original  function  y. 

(3.)  The  considerations  just  presented  analytically  admit  of 
geometrical  illustration.  For,  whatever  may  be  the  relation  be- 
tween X  and  y,  the  former  may  be  regarded  as  the  abscissa,  and 
the  latter  as  the  ordinate  of  a  plane  curve;  and  the  determination 
of  the  relation  between  the  corresponding  increments  of  x  and  y, 
is  reduced  to  finding  the  change  in  the  length  of  the  ordinate 
produced  by  an  arbitrary  change  in  the  length  of  the  abscissa. 


FIRST  PRINCIPLES.  17 

It  is  the  chief  object  of  the  Differential  Calculus  to  investigate: 
the  laws  of  increase  of  functions  having  various  forms,  when  sucl 
changes  are  produced  by  an  arbitrary  change  in  the  value  of 
the  independent  variable  upon  which  the  values  of  the  functions 
depend. 

Geometrical  considerations  will  also  point  out  very  clearly  how 
it  happens  that  a  given  augmentation  of  the  variable  x  will,  in 
different  stages  of  its  magnitude,  produce  widely  different  increments 
of  the  function  y. 

Referring  to  the  an- 
nexed diagram,  it  will 
be  apparent  that  near 
the  vertex  G  of  the 
^urve  CPE^  a  slight 
increase  in  the  value  of 

the  abscissa  x  wmU  produce  a  comparatively  large  increase  in  the 
value  of  the  ordinate  y ;  but  when  the  tangent  to  the  curve  forms 
a  smaller  angle  with  the  axis  OX^  as  at  P,  the  same  increment 
in  X  will  produce  a  much  smaller  increase  of  y ;  and  if  the  tangent 
be  nearly  parallel  to  OX,  the  increment  received  by  y  will  be  very 
small  in  comparison  with  that  given  to  x.  Finally,  by  continuing 
to  increase  x^  the  ordinate  y  may  first  cease  to  increase,  and  may 
afterwards  actually  decrease,  or  the  increment  of  y  may  become 
negative;  and  these  different  results  will  occur  without  any  change 
in  the  form  of  the  function  y. 

4.  One  of  the  first  inquiries  presented  for  consideration  is  the 
determination  of  the  general  form  of  the  function  F{x  -f  A) ;  for, 
since  we  desire  to  compare 

y  —  Fx         with         y^  —  F{x  +  A), 
it  is  important  to  know  what  form  F{x  -f-  h)  will  assume  when  ex- 
panded  into  a  series  of  terms  involving  x  and  h.     Hence  the  fol-. 
lowing 


18  DIFFERENTIAL    CALCULUS. 

Fro2)ositlon.  To  determine  the  general  form  of  the  development 
of  any  function  of  the  algebraic  sum  of  two  quantities,  such  as 
F(x  -f  ^Oj  arranged  according  to  the  powers  of  the  second  h. 

1st.  There  must  be  one  term  m  the  development  of  the  form 
Fx.  and  the  other  terms  must  contain  h.  For,  since  the  develop, 
ment  is  supposed  to  be  general,  and  therefore  true  for  all  values  h^ 
it  ought  to  be  applicable  when  h  =  0,  in  which  case  the  undeveloped 
function  F(x  +  h)  reduces  to  Fx.  This  condition  wmII  be  satisfied 
by  supposing  the  first  term  in  the  development  to  be  Fx,  and  all 
Jhe  succeeding  terms  to  contain  powers  of  h,  since  the  supposition 
^  =  0  will  then  give  rise  to  an  equation,  Fx  —  Fx,  which  is  identi- 
eally  true.  And  no  other  conceivable  form  of  development  would 
lead  to  this  result.  ''^ 

.  We  may  therefore  write 

F(x  +  k)  =:  Fx -\- Ah" -{- Bh^  +  Ch' -{- 6zc.         (1), 

in  which  the  coeflicients  A,  B,  C,  &c.,  will  usually  be  functions  of  ar, 
and  the  exponents  a,  6,  c,  (Sec,  undetermined  constants. 

2d.  None  of  the  exponents,  a,  b,  c,  &c.,  can  be  negative.  For  if 
there  could  be  a  term  of  the  form 

Bh-^        or         y,, 

it  would  become  infinite  when  k  =  0,  thus  rendering  the  developed 
expression  infinite,  while  the  undeveloped  expression  would  become 
simply  Fx,  &nd  this  latter  would  probably  be  finite. 
/  3d.  None  of  the  exponents  can  be  fractional.     For  if  there  could 
be  a  term  of  the  form 


Eh'         or         JS'/h' 


such  term  would  have  as  many  different  values  as  there  are  units 
in  s ;    that  is,  it  would  have  «  values ;    and  each  of  these  val\ie9 


FIRST  PRINCIPLES.  19 

could  be  combined  in  succession  with  the  aggregate  of  the  othdr 
terms  of  the  series. 

Now  if  each  of  these  other  terms,  except  the  first  term  Fx^  be 
supposed  to  have  but  one  value,  the  sum  of  all  the  terms  containing 
h  will  have  s  different  values.  And  if  Fx  be  susceptible  of  n  dif- 
ferent  values,  the  entire  development  will  admit  of  ti  X  s  values, 
since  each  value  of  Fx  may  be  combined,  in  succession,  with  each 
value  of  the  remaining  terms. 

But  F{x  -f-  /*)  being  of  the  same  form  with  Fx^  must  have  the 
same  number  n  of  values.     Thus,  for  example,  if 

F{x-\r'ti)  =  {x^-hy,        then        Fx  =  (^, 
and  both  will  have  three  values. 

If  F(x^K)^a{x-\-hY^-h(x-\-hy, 

then  Fx  z:z  ax^ -\- h3^  ^ 

and  both  will  have  five  values,  &c. 

Thus,  in  the  case  supposed  above,  where  there  was  one  fractional 
exponent,  F{x  +  A)  would  have  n  values  when  undeveloped,  but 
»  X  s  values  when  developed— a  manifest  absurdity. 

We  conclude  therefore  that  the  exponents  cr,  6,  c,  &;c.,  in  the 
general  development,  must  be  positive  integers ;  and  in  order  to 
nfiake  the  development  include  every  possible  case,  we  write 

F{x  ■\-}i)-Fx-\-  Ah  +  Bh^  +  Ch?  -f  Bh^,  &c., 

including  every  power  oi'h.  If  in  any  particular  case  some  of  these 
terms  should  be  unnecessary,  it  will  suffice  to  suppose  the  cor- 
responding coefficients  A^  B,  (7,  &c.,  to  reduce  to  zero. 

We  have  a  familiar  example  of  the  expansion  of  F{x  +  h)  in  the 
well  known  binomial  theorem.     Thus,  if 

F(x -\- h)  —  (x  +  hy  =  x"" -{- nx^'-Vi 
nin  —  \)         „,„       w(w  — 1)(«  — 2) 


20  DIFFERENTIAL    CALCULUS. 

we  shall  have 

^ere  A^  B^  (7,  &c.,  are  functions  of  x. 

The  following  are  likewise  examples  of  the  development  as  apt 
plied  to  particular  cases. 

2.  Let  Fx—  {a'\-  xy+  bx^ :     then 

F{x-\-J^  =  {a  +  x  +  hy  +  b(x  +  A)«, 

^hich  expressions,  when  expanded  by  the  binomial  theorem,  give 

F{z  +  h)  =  (a  +  x)^-\-^(a  -{-  x)'^  h  -^{a  -^  xf*  h^  +  &c., 

+  bx*  +  nbx"'^ h  -f  -^\  ~^^  ^a;'»-2A2  +  &c. 

=  i^a:  +  rw6a;«-i  +  «  ("  +  ^)~  J ^ 

^hich  corresponds  with  the  general  form, 
3    Let  Fx  =  log  X :     then 

F(x  +  h)  =  log  (.r  +  A)  =  log  [x  (l  +  ^)]  =z  log  a:  -h  log^'l  +  ^) 
f^here  Jlf  denotes  the  modulus  of  the  system  of  logarithms, 
frhich  also  corresponds  to  the  general  form. 


FIRST   PRINCIPLES.  21 

It  may  be  well  to  observe,  that  although  the  form  of  the  develop- 
ment of  F{x  -f  Jf)  is  always  sach  as  has  been  indicated  while  a; 
retains  its  general  value,  yet  it  is  possible  (in  some  cases)  to  assign 
certain  particular  values  to  x  which  shall  cause  the  development 
in  this  form  to  become  impossible. 

Thus,  if  in  the  second  of  the  above  examples,  we  put  x  —  ^  a, 
the  true  development  of  F{x  +  h)  will  become  einiply 

F{x-\rh)=l^  -\-b{-  ay  -f  bn{-  «)«•->  h  -f  &c., 

in  which  one  fractional  exponent  appears. 

The  same  supposition  causes  all  the  coefficients  involving  negative 
powers  of  a  4-  a:  to  become  infinite  in  the  general  expansion.  If 
will  be  shown  hereafter  that  the  particular  cases  in  which  the 
general  development  is  inapplicable,  are  always  indicated  by  sonne 
of  the  terms  of  the  development  becoming  infinite.  At  present 
it  is  sufficient  to  remark  that  the  number  of  such  cases  is  compara 
tively  small,  and  that  they  will  receive  a  special  examination. 

6.  From  the  development  of  F(x  -f  A),  we  derive  a  direct  and 
general  method  of  finding  the  differential  of  any  proposed  function 

y  =  Fx, 

For,  if  we  give  to  x  an  increment  h,  we  shall  have 

yi  =  F{x  +  h)  =:Fx-{-  Ah-\-  Bh"^  +  Ch^  +  &c. 

r  .y^  —  y  =1  F{x  +  h)  -  Fx  =:  Ah  -\-  Bh^  +  Ch^  -f-  &c. 

. . . yjLZj.  ^A-^Bh^Ch?'-^-  &c. 
h 

And  by  passing  to  the  limit,  when  A  =  0,  we  get 

-f-  =  ^,        whence        dy  =  Adx, 
ax 


22.  DIFFERENTIAL   CALCULUS. 

Thus  it  appears  that  the  coefficient  A  of  the  1st  power  of  h 
in  the  development  of  F(x  4-  h)  is  the  differential  coefficient  of 
the  proposed  function,  and  this  multiplied  by  dx  gives  the  re- 
quired differential  of  y. 

It  will  be  found  convenient,  however,  to  form  rules  for  dif- 
ferentiating  functions  of  the  various  forms  likely  to  arise,  and 
to  this  investigation  we  proceed  next. 


/ 


CHAPTER    II. 

DIFFERENTIATION    OF    ALGEBRAIC    FUNCTIONS. 

6.  Prop.  To  differentiate  the  product  of  two  functions  of  a  sin- 
gle  variable. 

Let  u  =  yz, 

^here  y  and  z  are  given  functions  of  the  same  independent  variable 
x^  and  let  x  take  an  increment  A,  converting  u,  y,  and  s,  into  Wj,  yj^ 
and  2i.  Then,  since  y-^  and  z-^  will  each  be  a  function  of  a:  -j-  K 
we  shall  have 

and  2i  =  0  +  ^jA  4-  ^^^2  _}_  c^^^  _j_  ^c. 

•     .  • .  t^i  =  yi^i  =  y0  +  (^s  +  ^liy)A  +  {Bz  +  ^^y  +  AA;)h^ 
-f  (6^2;  4-  Cjy  +AB^  +  ^i^)^^  +  &c. 

.  • .  ^^  =  J^fip^  ^Azi-  A,y  +  (i?.  +  B,y  -^  AA,)h 

+  (Cs  +  Ciy  +  ^A  +  Ji^)A2  4-  &(i 
and  when  A  =  0,  this  becomes 


du 
dx' 

=  Az  +  A^y.  = 

dy           dz 
''dx'^^dz' 

y, 

since 

A 

dy 
~  dx 

and 

A-- 

dz 
~  dx' 

And 

by  multiplying 

by  dx,  we  get 

du 

=  edy  4-  ydz. 

24  DIFFERENTIAL   CALCULIS. 

Thus  the  differential  of  the  product  yz  of  two  functions  is  found  by 
multiphjing  each  function  by  the  differential  of  the  other  function^  and 
adding  the  results. 

7.  Prop.  To  differentiate  the  product  of  several  functions  of  a 
single  variable. 

1st.  Let  u  —  vyz,  where  v,  y,  and  2,  are  functions  of  the  inde- 
pendent variable  x. 

Put  yz  =  s ;         then         u  =  vs, 

and  by  the  last  proposition, 

du  =  vds  -{-sdv^        and  also         ds  =  ydz  +  zdy. 

Substituting  the  values  of  s  and  ds  in  that  of  du,  there  result* 

du  =  V  [ydz  +  zdy)  +  yzdv  =  vydz  •\-  vzdy  -f-  yzdv. 

2d.  Let  u  =  svyz. 

Put  yz  =:  w ;         then         u  =  svwy 

.  • .  du  =  svdw  -f-  swdv  -}-  t't^cfs  =  sv(ydz  +  2:cfy)  4  syzdv  -f  fy?«^s, 

or,  ifw  =  svydz  -f-  5^'^rfy  +  syzdv  +  vy^rf.?  ; 

and  the  same  method  could  be  applied  to  the  product  of  a  greater 
number  of  functions. 

Hence  w^e  have  the  following  rule  for  the  differential  of  the 
product  of  several  functions  : 

Multiply  the  differential  of  each  factor  by  the  continued  product  of 
all  the  other  factors,  and  add  the  results. 

8.  Prop.  To  differentiate  a  fraction  whose  numerator  and  denom. 
inator  are  functions  of  a  single  variable. 

Let  u  =  -,  where  y  and  z  are  functions  of  x, 

z 


ALGEBRAIC  FUNCTIONS.  25 

Then  y  =  mz,  and  this  differentiated  by  the  rule  for  products, 
givea 

y 

dv  =  udz  -f  zdu  z=z-  -dz  -\-  zdu 
z 

. ' .  zdy  z=:  ydz  +  z'^dn^ 

11-1.  .         ^^y  —  y^^; 

and  by  reduction  du  =  — '■ — 

z^ 

Thus  the  rule  is  as  follows : 

Multiply  the  differential  of  the  numerator  by  the  denominator^  and 
the  differential  of  the  denominator  by  the  numerator ;  subtract  the 
second  product  from  the  first,  and  divide  the  remainder  by  the  square 
of  the  denominator. 

9.  Prop.  To  differentiate  a  power  of  a  single  variable. 

1st.  Let  u  z=z  ic",  where  n  is  a  positive  integer. 

Regarding  x"*  as  the  product    x.  x,  x.  ar,  &c.,  of  n  equal  factors 

each  =  ar,  and  applying  the  rule  for  differentiating  a  product,  we 

get 

du  z=  x'^-^dx  +  x'*-'^dx  -f  x^^-^dx  -f  &c.,  to  n  terms. 

.  • .  du  ■=.  nx^-'^dXj 

and  the  rule  in  this  case  is  the  follownig : 

Multiply  the  given  power  (x")  by  the  exponent  (giving  nx")  ;  then 

diminish  the  exponent  by  unity  (giving  nx"""^)  ;  and  finally,  multiply 

by  the  differential  of  the  root  (producing  nx°~^dx). 

2d.  Now  suppose  the  exponent  n  to  be  a  positive  fraction  - 

c 

« 

Then  u  =  x~ 

,  • .  u^  =  a:«,  where  the  exponents  a  and  c  are  both  positive  integers. 

Hence,  by  the  application  of  the  rule  just  established  for  such 

cases,  we  have 

cu^^^du  =  ax'^^^dx 

,        aa;«-i  -        a      a;"-'      ,        a   tt-i-.o4  -  ,        a   «— i  , 
, • .  du  = dx  = ; — J dx=  -X        ^  «dx  r=z-X^     dx. 


(/)'-' 


26  DIFFERENTIAL   CALCULUS. 

and  the  rule  for  differentiating  thp  power  is  the  same  as  when  the 
exponent  is  a  positive  integer. 

3d.  Let  the  exponent  be  a  negative  integer,  or  w  =  tt"^ 

\  X 


Then  w  =  —  = 


n+l 


and  this  differentiated  by  the  rule  for  fractions,  gives 

x^'ri  _  (;i  +  l)a:"+^                  ndx  _  ,  , 

du  = \  .  ^    ^ dx= -TT  =  —  nr-^-^dx. 

And  the  rule  is  still  the  same. 

4th.  Let  the  exponent  be  a  negative  fraction,  or  let  u  =  x 
Then  u"  =  a^*,  and  by  the  first  and  third  cases, 


cu^-^du  =  —  aoir'^^dx,        or,        du  = dx. 

a     xr'^^dx            a  -?-i  . 
du  = ; ^ = X   '      dx. 


'     (.-)- 


and  the  formula  is  still  the  same. 

We  might  have  deduced  the  rule  for  differentiating  a  power,  as 
alike  applicable  to  all  cases,  by  employing  the  binomial  theorem ; 
for,  since  the  second  term  in  the  development  of  (aj  +  A)",  is 
na:*»-'A,  for  all  values  of  w, 

we  must  have  — ^ — -  =  na:"-',     or,     d  (a:")  =  nx^-^dx. 

dx  '         '        \     / 

It  is  intended,  however,  to  demonstrate  the  truth  of  the  binomial 
theorem  by  the  aid  of  the  differential  calculus,  and  hence  the  neces- 
sity of  establishing  the  rules  for  differentiation,  without  reference  to 
that  theorem. 

Remark.  If  the  function  which  it  is  proposed  to  differentiate 
contain  a  constant  factor,  such  factor  will  appear  in  the  differential. 


ALGEBRAIC  FUNCTIONS.  27 

Thus  d  {ax)  =  adx,  for  when  x  takes  the  increment  A,  the  function 
ax  becomes 

/       I     l,\  A  U-i  —  U  J         ^^ 

«,  =  a  (a;  -f  A)      and      .  • .  ~~ —  =  a      and      -^  =  a. 
^         ^  '  Ji  dx 

Similarly  if  m  =  a .  Fx,  where  F  denotes  any  function, 

then  u^  =  aF  (x  +  h)       and       du  z=z  ad  [Fx) . 

10.  Prop.  To  differentiate  the  algebraic  sum  of  several  functions* 
of  a  single  variable. 

Let  u=:As-{-Bv—Ci/-h  Dz, 

where  5,  v,  y,  and  z,  are  functions  of  ar. 

Then  when  x  takes  the  increment  h, 

As  becomes  As^  z=  A  [s  -\-  A^h  -\-  B-Ji?  -f-  C^"^  &c.). 

Bv  becomes  Bvy^  =  B  {y  ^-  AJi -\-  Bji^  +  CJi^  &lq,), 

Cy  becomes  Cy^  =  C  {y  +  A^h -\-  BJi^  +  CJi^  &c.). 

Dz  becomes  Dz^  =  D  {z -{-  A^h  +  B^h^  +  C4A*  &;c.). 

,  *.  u  becomes  u^  =  As  -^  Bv  —  Cy  -{-  Dz 

+  {AA^  4-  BA^  -  CJ3  -{-DA^  h  +  &;c. 

.  • .  du-—  {AA^  +  BA^  —  CA^  +  DA^)  dx. 
But       A-^dx  =  ds,       Azdx  =  dv,       A^dx  =  c/y,       A^dx  =:  dz. 

.'.  du  =  Ads+  Bdv  —  Cdy  +  Ddz, 
And  the  rule  is  as  follows : 

Differentiate  the  terms  successively^  and  take  the  algebraic  sum  of 
the  result. 

'  Remark.  If  a  constant  be  connected  with  a  variable  quantity  by 
tne  sign  -f  or  — ,  such  constant  will  disappear  by  differentiation. 
Thus,  when  we  have  u  =:  a  -{•  Fx^  then 

u^=i  a  -{-  F  {x  •{-  h)  :=z  a  -{■  Fx  -^  Ah  -{-  Bh"^,  &c., 

=  w  -f-  Ah  4-  ^A^  &c. 

.  • .  rfw  =  Adx,  the  constant  a  having  disappeared. 


28  DIFFERENTIAL  CALCULUS. 

EXAMPLES. 

'11.    1.  To  ditferoritiate 

//  =  4^3  4.  7^2  _  8^  4-  5, 
Applying  the  rule  tor  powers  to  each  term  we  obtain 

dy^'iX  ^5  x^dx  -\-l  X  2xdx  —  Sdx  =  ( 1  '^x^  -f  I  Lc  -  S)dx. 

.'.-^  =  12a:2-j-  14^-8. 
ax 

2.  1/  =  ax^(bx  -\-  c)  =:  abx^  -f-  uvx^. 

Differentiating  this  as  a  product,  we  get 

dy  =.  2ux[bx  -f-  c)dx  +  ax%dx  =z  {l^ahx"^  -\-  2acx)dx. 

Or  by  first  perfomiing  the  multiplication  indicated,  and  then  dif 
ferentiating  as  a  sum,  ihe  same  result  is  obtaint-d. 

r.^  =  Sabx^-\-2acx. 
dx 

Differentiating  by  the  rules  for  fractions  and   powers,   we  obtain 
VZx'\b  +  x'^fdx  —  8(6  -f-  x^f  X  Ax^  x  2xdx 


dy  = 


{b  +  ^2)6 


l'>x%b  -}-  x^)  -  2ix*  ^         12.f2(/>  -  .r^)    , 
= ^^ dx  = ax. 

{b   -h  X^y  (b     f   ^2)4 

dy  _  1 2^:2(6-  x'') 
'    '  lx~  '~{b~+x^y    ' 

y  =  -/a  +  bx"^  =  {a  -f-  bx'^)^, 
2  '  dx       ^a  -f  bx^ 


ALGEBRAIC   FUNCTIONS. 

5.  u  =  x{l  +  x^)(l  +  x^). 

du 

—  =  (1  +  a:2)(l  -h  x^)  +  x{\  +  x^)  X  2a;  +  x{\  +  a;^)  x  3«« 

=  1  +  ic2  4-  ar3  +  a;5  +  2^2  ^.  2a;5  +  3a;3  +  3a:« 

=  1  +  3a;2  +  4a;3  +  62:*. 

6.  M  =  yr+'/nf^  ==  [a;  +  (1  +  a;2)*]* 


29 


-{x+^l^-x^) 


_'/a;+v/M-a;2 

V^+ yT+lc2  X y^FT^         2-/!  H-^2 


7. 


w  :=  6a; 


8.   «  =  —  —  6  =  car®  —  b. 


du  8  * 
-—  =  -  bx  , 
dx       3 

du 
ax 


6c 


u  =  V^Vv^  + 1  =  «  (^  +  1)  • 

eft*  =  lx'*{x^  +  1  )*c?a;  +  I  x^{x^  +  1)"*  X  I  x'^dx. 

O  ,6  <^ 


du  _  {x^  +  1)* 


*    dx 


Sx 


+ 


7a;* +  4 


4a: 


(x^+l)^       12V5\4^+1 


10. 


_  Vi  +a;4-  VT-x  _  (-y/l  +a:+  y^l  -a;)' 
~  2a; 


,^-\-  X  -  yT- 


1  +  VI 


«fw  _  -  x^(\  -  x^f^  -  (1  4-  -y/T^  a;2)  _  _  1  +  -/l  -  a;^ 


dx 

11.    «  = 


ar2v'i 


ic  +  ■v/r+  a;2 


=  -(--^/^^^)=.vT+l^- 


(1  +  z') 


30  DIFFERENTIAL    CALCULUS. 


12.  u  =  y^pT-i-  +  Y^ZT^Y 

3^  4.1; 


2x-x/x      yc^  -  x^ 


V' 


^x 


13.    t«  =  W  «  +  .^  +  V  a  +  a;  H-  -y/a  4-  a;  &c.,  continued  indefl 
nitely. 


Here         «  =  -\A~H-  ar  +  «*,         and         .  • .  w^  =  a  +  ^  +  w, 
or,         I 

c/m.  1 


u  =  a  +  x,  ,-.%  =  -  +  y/a  4-  a^  -I-  ^, 


dx 


y4:a  -h  4a;  +  1 


The  functions  considered  hitherto  are  called  algebraic  functions, 
because  they  require  only  the  performance  of  the  common  algebraic 
operations  of  addition,  subtraction,  multiplication,  division,  raising 
of  powers,  and  extraction  of  roots.  There  is  a  second  and  very 
extensive  class  of  functions,  in  which  the  variable  enters  as  an 
exponent,  or  in  connection  with  logarithms,  sines,  cosines,  tangents, 
circular  arcs,  &;c.,  of  which  the  following  are  examples :  a',  x*, 
logo:,  sin  a:,  (cosar)"'"**,  sin-^  ar,  (loga:)^^"^,  &;c.  These  are  called 
transcendental  functions,  and  they  will  be  considered  m  the  next 
chapter. 


or 


i^  I  li  K  A  It  V   '^ 
k   CAIJFoi^xi  Y 

CHAPTER    III. 

TRANSCENDENTAL   FUNCTIONS. 

12.  Prop.  To 'differentiate  w  =  log  a;. 

Let  X  take  the  increment  A,  converting  u  into 

Wj  =  log  (a;  +  h). 
Then  «i  =log  (  a;  +  A)  =  log  ^  (l  +  ^H  ==  log  ar  -f  log  (l  f  -V 
_L  ir/^        A2         A3         A*   .      \ 


where  M  is  the  modulus  of  the  system. 

,  • .  —  =    ^  ,      '  =  —         and         G?w  =  —  dx. 

dx  dx  X  X 

Hence  the  rule  is  as  follows : 

Multiply  the  differential  of  the  variable  by  the  modulus  of  the  sys 
tern  in  which  the  logarithm  is  taken^  and  divide  the  product  by  the 
variable. 

If  the  logarithms  belong  to  the  Naperian  system  whose  njodulus 
is  equal  to  unity,  we  shall  have 

(/(log  2;)=.-^. 

As  the  essential  properties  of  logarithms  are  the  same  in  all  sys- 
tems, while  the  form  of  the  differential  is  simplest  in  the  Naperian 
system,   the  logarithms    employed    throughout    the    Calculus    will 


82  DIFFERENTIAL    CALCULUS. 

always  be  the  Naperian,  unless  the  contrary  is  distinctly  specified, 
and  the  rule  for  differentivating  a  logarithm  will  be  simply  this: 

Divide  the  differential  of  the  quantity  by  the  quantity  itself. 

13.  Prop.  To  differentiate  an  exponential  function  as  u  =  a',  the 
base  a  being  constant. 

Passing  to  logarithms  we  have 

log  u  =:  X  log  a. 

du 
.  • .  c?(log  u)  =  d{x  log  a)         or         —r-=zioga.dx', 

',  du  =  log  a.u.dx  z=:  log  a .  a*,  dx        and         —  =  log  a  .  a*. 

And  the  rule  for  differentiating  an  exponential  is  this : 
Multiply  the  exponential  (a^)   by  the  differential  of  the  exponent 
(dx),  and  that  product  by  the  Naperian  logarithm  of  the  base  (log  a). 
Cor.  If  a  =  e,  the  Naperian  base,  we  shall  have  log  e  =  1 ; 


aiju 


.  • .  d{e')  =  e'dx,        and         -~-^  =  e'. 


Remark.  The  rule  for  differentiating  logarithmic  functions  will 
often  be  found  useful,  even  when  the  original  function  is  algebraic, 
since  by  passing  to  logarithms  we  may  give  the  function  a  simpler 
form. 

Examples  af  Logarithmic  and  Exponential  Functions. 
14.  1.  Let  u  -  log  {x  -h-y/F-f"^. 


du  =  ^(^+v^+_^  ^ !  +  (!  +  ^_y^^ 

«  4-^/1  +a:2  ,  dx  du 

^  dx  =  '         — 


(•^-+Vl  +•'»') -A-t-*^  /rH-a:2    *    'dx     yTl-a;» 


TRANSCENDENTAL  FUNCTIONS. 


33 


Passing  to  logarithms  we  have 

log  w  =  log  a;  -f  log  (a2  ^  x^)  ■\-  -  log  (a^  —  x^). 


u~x  a?-\-x^     "^2    a2~a;2 


(/;c         2a;c?^ 


xdx 


X        a^  -^  x^       a^  —  a;2 


.!=(<.«  +.»)  vs^^^+  2.V"^  -  .^  -  ^^^;g 


r.2  _  9:2 


3. 


w  =  log 


v<^Mrr__ 


Multiplying  numerator  and  denominator  by  the  numerator  we  have 
2ar2  4-  1  -  2x  y/x^  -f-  1 


'^  =  i«g W:^i 


du      4x-  2-vAM-  1  -  2a;2(a;2  +  1) 


log  (2a;2  +  1  _  2a:'vAM-l) 


4.  w  =  a;«V-i.     Then  log  «  =  a-y/—  1  log  x, 

^    du  _       / — -  dx  ^: 

and  du  =  a^/—  1  .  .r«V^ .  —  =  a^/—  1  aj^v^-^  c?a:. 

V  a;         ^ 

Thus  the  rule  for  differentiating  a  power  is  still  the  same,  when  th« 
exponent  is  imaginary 


34  DIFFERENTIAL    CALCULUS. 

\    6.  w  =  X*.     Then  log  w  =  a;  log  x. 

.♦.  —  =z\o%x,dx  -\-  X, —  =  (log a;  -f  l)daf 

This  signifies  that  x  is  raised  to  a  power  whose  exponent  is  a:*, 
and  it  must  not  be  confounded  with  (a;*)',  which  latter  implies  that 
«*  is  raised  to  the  x*^  power. 

u  w  dx 

Thep     log  w  =  a:*  log  a:  .  • .  —  =  log  a;(logar  -f  l)a;*(/x  -f  ^*  — 

u  X 

.  • .  ^  =  X*' .  a;'  l^log  a;(log  a:  +  1 )  +-J 

7.  u  =  e**     where  e  is  the  Naperian  base. 

log  u  z=  X*  log  e  =z  X'    ,'.-—  =  e*',  x*  (log  x  -|-  1). 

8.  tt  =  ««*.  Then        log  w  =z  e*  log  a: 

du  xA  1\ 

...-  =  a:«(loga:  +  -j.'. 

0.  M  =  log  (nar) .         Then         du  =  — ^^ — ^  =  — . 

^^     '  nx  X 

This  result  is  the  same  as  when    u  =  log  ar,  as  might  have  been 
anticipated  since  log  {nx)  =  log  n  -f  log  a:,  and  log  n  is  constant. 

10.    »  =  log(log^).     Then    rf„  =  iil2Sf) .  =  _^ 

log  a;  x.loga; 

du  _      1 
'  dx""  z  log  X 


TRANSCENDENTAL  FUNCTIONS.  35 

11.     w  =  (log  xY  =  log  "ar.     Then  du  =  n  log  "-^  x .  c?(log  z) 

du      n  .  log  »-^  ar 
'   '  dx~  X         ' 


12.         «  =  e  log  V^H^.     Then     log  u  =  log  ^d^+x^ 
,',  u  =^a?  4-  a:2  =  (a2  +  a;^)*      and     ^  = -^ 


13.  u  =  c'''^''".  cZw  =  «'°'""  .  c^  (log-a:) 

.  • .  -7-  =  -  •  c         .  log'-^ar. 
dx       X  ° 

14.  "  =  7  ^  ^^g^^  ~  8  ^*  ^^^  ^  "*"  32  *** 

---  =  ic^  log2a;  +  -x^  log  a;  —  -  a;3  log  x  —  ~x^  -\--x^  =  3^  log^x. 
dx  Z  Z  o  o 


15.  u  =  e'(x^  -  4a;3  +  12a:2  _  24a;  +  24) 

^  =  e'{x*  -  4a;3  +  12a;2  -  24a:  +  24) 


+  e*{4x^  -  ]2a;2  4-  24a;  —  24)  =  e* .  ar*. 


Trigonometrical  JFunctions. 

15.  The  trigonometrical  functions  sin  a*,  cos  a;,  tan  x,  &c.  will  next 
be  considered,  but  the  determination  of  the  forms  of  their  differen- 
tials will  be  facilitated  by  the  following 

mi     1.    .  ,  .      arc     arc  .     arc  .         . 

Fi'op.   Ihe  limit  to  the  ratios  -r— ,  -; ,?    and    )     when  the 

sm    chord  tang 

arc  is  diminished  indefinitely,  is  unity. 

,  ^.  sin         cos         rad— versin       ,        versin 

Proof,  femce      =  — - —  = =1 — -, 

tan       radius  rad  rad 


36 


DIFFERENTIAL  CALCULUS. 


and  since  the  last  term  in  this  equalHy  can  be 

rendered  smaller  than  any  assignable  quantity 

by  taking  the  arc  sufficiently  small,  it  follows 

,     ,.    .  ,  .      sin    . 

that  the  limit  to  the  ratio  is  unity. 

tan 

But  both   the  chord  AB  and   the  arc  AB 

are   intermediate  in  value   between  the  sine 

BD  and  the  tangent  AT,      Hence  at  the  limit,  when  the  arc  ia 

indefinitely  small, 

arc 


arc 
sin 


arc 
tan 


sm 
tan 


chord 

16.  Prop.  To  differentiate    y  =  sin  x. 
to.  the  well  known  trigonometrical  formula, 


make 
Then 


sin  a  —  sin  6  =  2  sin  -  (a  —  b)  cos  -.{a  -{-b)^ 


\{a-b)=lh, 


x-\-  h        and 
and 


2 

b  =  x. 


\(a  +  b)  =x  +  ^h. 


.  • .  sin  (a;  +  A)  ~  sin  a;  =  2  sin  - h .  cos  {^  +  ^h). 


am  {x  -\-  h)  —  sin  x      2  sin  -  A .  cos  (a:  +  -  h) 


2 

1 
2 


But  at  the  limit  when  A  =  0, 


h 


— cos  (a:  +  -  A), 


sm-A 


1,      and      cos  (a?  +  -  A)  =  cos  a?. 

<6 


TRANSCENDENTAL  FUNCTIONS.  37 

.  • .  -^  =  -^—; — -  =  cos  ar,       and       c?(sm  a;)  =  cos  a; .  aa;. 
dx  dx 

17.  Prop.  To  differentiate     y  =  cos  ar. 
Here  y  z=  cos  a:  =  sin  I  -  '^  —  x\ 

where  -jt  =  semi-circumference  of  the  circle  whose  radius  =  1. 

.  •.  dy  z=z  c?sin  1-*  —  a:|  =  conJ-it  —  x\  •  cil-'"^  —  x\=:  —  sinxdz 

dy       d  cos  X 
'    '    dx~      dx      ~  ' 

the  negative  sign  prefixed  to  the  value  of  this  ratio  signifies  that  the 
cosine  decreases  as  the  arc  increases. 

18,  Prop.  To  differentiate     u  =  tan  x. 

, ,         .        ,  sin  a;       cos  a; .  fl?  sin  a:  —  sin  a; .  c?  cos  z 

du  =  a  (tan  x)=  d = ■ 

^  '  cos  X  cos^a; 

cos^a;  -f  sin^a;  .  dx  07 

= dx  =  — —  =  sec^a; .  dx. 

cos^a;  cos-^a? 

die       d  tan  x 


=.  sec'^x. 


dx  dx 

19.  Prop.  To  differentiate   u  =  cot  a:. 

du  =  fl?(cot  :fc)  =  dtanl-'je  —  x\=:  sec^l-ir  —  x\.  d  [-or'  —  x\ 


=.  —  cosec^a;. 

dx 

du       dci^tx 

-r-  =  — ; —  =  —  cosec^ar. 

dx          dx 

20.  Prop.  To  differentiate   w  =  sec  a;. 

-T  1  ,  1    1  —  d,  cos  X      sin  2; .  d» 

liere  u  =  sec  x  = •    ,',  du  =  d = = r — 

cos  a;  cos  a;  cos^a;  cos^a: 

^  ,  ,  du      d  so.cx 

or,  du  =  tan  x .  sec  x .  dx       and       .  * .  -7-  —  — ; —  =  tan  x .  sec* 

dx  dx 


38  DIFFERENTIAL  CALCULUS. 

21.  Prop.  To  difTerentiate   u  =  cosecar. 

du  =  </(cosec2;)  =  c?secj-'n'  —  x\ 


= 

—  cot  a; .  cosec 

xdx. 

d  cosec  a: 

-  cot  a; . 

,  cosec  X. 

22.  Prop.  To  differentiate    u  =  versin  a:. 

c?M  =  c?  (versin  x)  =  g?(1  —  cos  x)  =  sin  aria:, 
du       d  versin  x 

.  •  .     -r-  =  ; =  sni  X. 

ax  ax 

23.  Prop.  To  differentiate   u  =  coversin  x. 

du  =z  c?  (coversin  a:)  =  i.versinl-'B'  —  x\  =  sin  I-*  —  a'lif-'n-  —  x\ 

-  du        d  coversin  x 

z=  —  cosx.dx.  ,'.-—=  , =  —  cos  X. 

ax  ax 

24.  In  each  of  these  expressions,  x  represents  the  length  of  an 
arc  described  with  a  radius  equal  to  unity,  and  the  radius  does  not 
appear  in  the  formulae  :  but  it  is  necessary  to  remember  that,  in 
each  case,  ^  =  1  must  be  understood  to  enter  into  the  formula  as 
often  as  may  be  required  to  make  the  two  members  of  the  equation 
homogeneous. 

Geometricfiil  Illustration. 

25.  The  results  just  obtained  may  be  illustrated  geometrically  in 
Buch  a  manner  as  to  convey  a  more  precise  view  of  the  compara- 
tive small  changes  imparted  to  the  several  trigonometrical  functions, 
hy  an  arbitrary  small  change  in  the  arc  upon  which  they  depend. 


TRANSCENDENTAL  FUNCTIONS. 


39 


Thus  let  ab  represent  an  arc  x  described 
with  rad  =  1 ,  and  bh^  =  dx  sl  small  in- 
crement given  to  x.  Then 
eb  —  sin  x,  ce  =  cos  ar,  at  =  tan  x,  ct  =  sec  ar, 
sb^  =  d .  sin  x^  sb  =  d .  cos  x,  iti  =  d .  tan  a;, 
r^i  =  0? .  sec  a:. 

Also  when  bb-^  is  diminished  continually, 
the  small  figures  bsb^  and  trt^  will  continu- 
ally approach  to  the  forms  of  right  angled  triangles,  becoming  ia 
definitely  near  to  such  forms  at  the  limit.  Moreover,  the  two  small 
triangles  will  then  be  similar  to  cbe.  Hence  we  shall  have  the 
proportions 

cb  :  ce  : :  bb^  :  b^s    or     1  :  cos  x  : :  dx  :  rf  sin  ar  =  cos  xdx. 

cb  :  eb  : :  bb-^  :  bs     or     1  :  sin  x  :  :  dx  :  dcos  x  =  sin  xdx. 

The  latter  result  should  be  written     c?  cos  a;  =  —  sin  a: .  rfar,  be- 
cause the  cosine  diminishes  as  the  arc  increases. 

Again  we  have  the  proportions 

ca  :  ct  :  :  rt  :  tt^     )    .  ' .  ca  X  cb  :   {ct)^  : :  bb^  :  t\ 


and      cb  :  ct  '.:  bb-^  :  rt    )    or   P  :  sec^a;  \\  dx  \  d  tan  x. 


Also  ca  :  at 
cb  :  ct 


rt 


.  d  tan  X  =  sec'^xdx. 
rt^    )    . ' .  ca  X  cb  :  at  X  ct  :  :  bb^  :  rt^ 


bb^  :  rt   )    or   1^  :  tana:,  sec ar  : :  dx  :  d  secx. 
.' .  d  sec  X  =  tan  x .  sec  x  .  dx. 
In  the   same   manner,  expressions   for    rfcotar,    rfcoseca;,    &c.^ 
could  be  obtained. 

Circular  Functions. 

26.  We  will  now  consider  the  circular  functions,  sin~^a:,  tan-^a:^ 
&c.,  which  expressions  are  read,  the  arc  whuse  sine  is  ar,  the  arc 
whose  tangent  is  ar,  &c. 


4t)  DIFFERENTIAL    CALCULUS. 

In  these  cases,  it  is  the  arc  which  is  the  function,  or  dependent 
variable,  the  independent  variable  being  the  sine,  or  the  tangent,  &;c. 

27.  Prop.  To  differentiate   y  =  sin—^j?. 

Since  this  notation  is  intended  to  imply  that  y  is  the  arc  whose 
sine  is  equal  to  a;,  we  must  have  as  an  equivalent  relation 


dx  =  cosy  .  dy      and 


X  =■  ^my 
dy  1  1 


dx       cos  y       ^i  _  sin2y       yT"~  x* 
d  sin— ^ar  1 


dx 


,/r- 


X' 


28.  Prop.  To  differentiate  y  =  cos-^ar. 

Here  a:  =  cos  y,  .'.  dx  =  —  sin  y .  dy 

,    dy_ 1_ 1^ 1_ 

'   '  dx'~       sin  y  ~       ^1  -  cos^  ~  ~  ^^ 

d  cos~^jr  1 

dx     ~     yT"-^r^* 

29.  Prop.  To  differentiate   u  =  tan-^a:. 

X  =  tan  M,  .'.  dx  ^=  sec^w .  du 

,    du_  Jl_  _         1  _       1 

'    '  dx~~  sec^w  ""  1  4-  tan^it  ~  i  -\-  x^ 

d  tan-^r  _        1 
*    •        d7~  ~  1  -f  a:2* 

30.  Prop.  To  differentiate   u  =  cot-\r. 

X  =  cot  w,  .  • .  c?a;  =  —  cosec^w .  du 

du  I  1  1 


dx  cosec^w  1  -f  cot^w  1  -f  ^'^ 

d  cot""'x  1 

*   '        dx       ~  ~"  1  4-a:2  * 


TRANSCENDENTAL  FUNCTIONS.  41 

31.  Prop.  To  differentiate   u  =  sec-^a:. 

X  =  sec  M,  ,*.  dx  :=  tan  w  .  sec  w  .  du 

du  I  1  1 


dx       tan  w  .  sec  u       sec  ?^  y^ec^^—  1       x-^x^  —  1 
c?  sec~^a;  1 


fl^;c 


■y^ 


32.  Pro;?.  To  differentiate   u  z=  cosecr'^x. 


X  =  cosec  u,  ,'.  dx  z=  —  cot  u .  cosec  w  .  du 

du  1  1 


dx  cot  w  .  CDsec  u  ^osec  wycosec^w  ~  1 

1 


X-y/^-  1 

</  cosec-^ar  1 


d^  xy/x^  —  1 

33.  Prop.  To  differentiate   w  =  versin-^ar. 
z  =  versin  u     .'.  dx  =  sin  u  .du  =  -y/'l  versin  u  —  versin^w    du 
du  1  1 


^^       y^~versin  i^  —  versin^w       -^^Ix  — 
(?  versin—^a;  1 


or, 


rfa;  y^5^^  a;2 

34.  Projo.  To  differentiate   u  =  coversin-^a;. 

X  —  coversin  u 
.  • ,    dx  z=.  —  cos  u  .  du  =  —  y'2  coversin  u  —  coversin^z^    cfo* 

du_ 1 __  1 

<**  -y/^  coversin  u  —  coversin^u  -^^x  —  x^ 

d  coversin~^a?  1 

or,  ,  — —  =  —  -——^ • . 

dx  ^Azx  —  x"^ 


42  DIFFERENTIAL  CALCULUS. 

35.  The  differentiation  of  trigonometrical  and  circular  functions 
will  now  be  illustrated  by  examples. 

\  EXAMPLES. 

1.  u  =  S  sin*a;. 

(/m  =  3  X  4  sin^a; .  c?  sin  re  =  12  sin^rc .  cos  x .  dx 

du 
.  • .  -r-  =  12  sin^a; .  cos  x. 
dx 

2.  w  =  cos  nx. 

du  =  —  sin  nx .  d{nx)  =  —  nsmnx .  dx 

du 
.  • .  T"  =  —  w-  sm  nx. 
dx 

3.  u  =:  tan*«ar. 

du  z=  n  tan"~^  nx .  d  tan  nx  =  n^  tan"-^  nx .  sec^nar .  dx 

.  • .  -—  =  w^  tan"-^  nx .  sec^wa;. 
dx 

4.  u  =  sin  Bar .  cos  2a:. 

du  =  (3  cos  3a; .  cos  2a;  —  2  sin  3a; .  sin  2x)fl?a: 

du 
.  • .  -r-  =  3  cos  3a; . cos  2a;  —  2  sin  3a;. sin  2a;  =  cos  3a;  cos  2a;  -f-  2  cos  5ar. 
dx 

5.  u  z=z  (sin  x)*.      Then      log  u  z=z  x .  log  (sin  x) 

diL  du 

.• .  —  =r  [log (sm  a;) 4-  a;  cot a;](/a;  .•.—-=:  (sin  x)*.  [log  (sin  x)-\-  x  cot  x\ 

6.  w  =  (cos  a;)"'*-  *.     Then     log  w  —  sin  :t  log  (cos  x) 

.  • ,  —  =  (cos  a;)»'°  *  [cos  x  log  (cos  x)  —  sin  a;  tan  x\ 

7.  w  =  sin  (cos  a*),     c?m  =  cos  (cos  a:)fl? cos  x, 

du  •  f         \ 

.  • .  -r-  z=.  —  sm  a; .  cos  (cos  x), 
dx  ^ 


,9. 


Q/^V^ 


TRANSCENDENTAL   FUNCTIONS. 


43 


,aA.CL 


yi-h^' 


u  ■  — ■ 


V  1  +•  ^'      _ 


V 


(1  -I-  x'Y  -  a;2  (1  +  a:2) 


(i  +  ^^)(i-TTi^) 


-i        X'^^^'Y  ^*^ 


cfar 


9. 


10.  u  =  log 


i  +a;2 

c?M  _       1 
'  * '  ^  ~  1  +  x^' 

u  =  log  tan  X. 
du      ^<:,''x  1  _      2 

dx  ~"  taTi  t  ~ '  sin  a; .  cos  x  ~  sin  2a; 


1  +  sia  i       1  ,       ,    ,     .      \       1 T      /, 

^— -^^  -.    -  log  (1  4-  sni  x)--  log  (1  -  sin  ar> 


du 
dx 


_  1  r    cos  X  ocw?  ^    1  _       ^^^  ^      __     ■'^ 

~  2  Ll  4-  sin  a;       i  —  sm  x  J  ~~  1  —  sin^a:  "  cos  x 


IL 


12. 


w  =  sin~^  (S.^  —  4a;2). 
3  -  12^2  3 


-/I  -  (^x  -  4x''y-    -/i  - 

u  =  log  (cos  a;  +  \/—  1  •  sin  ^). 
<fM       -^—  1  .  cos  a;  —  sin  x         i — 


dx 


13. 


cos  a: 


1 


+-/=nr: 


1. 


du  = 


1 


-1  /^  ~f"  ^  •  ^^^  ^\ 
\a  +  6 .  cos  a:/ 

Ih  -\-  a.  cos  a:\ 
\a  -|-  6  .  cos  a:/ 
Y^a^  —  IP-      f.        ib  +  «  •  cos  ar\2 
V  \a  -{-  b  .  cos  a*/ 

a  sinar  (a  4-  &  cos  a:)  —  b^\nx{b  -\-  a  cos  a;) 
(a2  _  irf  {a-\-b  cos  ar)[(a  +  b  cosa;)2  _  (5  4.  «  cos  xYy 


dx' 


44  DIFFERENTIAL    CALCULUS. 

du  (a^  —  62^  sin  ^ 

^^       (a?  -  P)\a  +  b  cos  x)[{a^  -  P){1  -  cos^a:)]* 
1 


a  -i-  b  cos  X 
14.  u  =z  e"  cos  X. 

du  .  /  .       ^ 

— -  =  €*  cos  a;  —  c'^  sin  x  =z  e''  (cos  ic  —  sm  x). 
dx 


15.  w  =  tan-i  (yi  +  a;2  _  ^r). 

^  _        (1   -h  ar2)"^;g  ~     1  _  _  1 

d^~   \   +    (VTT^2  _  ^.)2  ~    .      2(1+  X') 


16.  w  =  log  Vsin  ^  +  log  ^cos  x. 

du       1  /cos  a;        sin  a:\  1 

rfo;  ~~  2  \  sin  ic        cos  x) 


sin  ic        cos  a-7       tan  'Zx 


17.  t^  =  log  -v  / h  -  tan-i  a; . 

V    X   —  X         4i 


-  log  (1  +  ;r)  -  -  log  (1  -  a:)  +  2  tan-»  ar. 


^  _         1  1^ 1  _       1 

<jfa;  ~  4(1  4-  a:)  "^  4(1  -  ar)       2(1  +  a;2)  "  1  -  a;** 

_  _                                     e^'  (a  sin  a;  —  cos  x) 
18.  «  = 5^ — „  — ^. 

^W  1  r  /         .  V       .  ■  .  1 

-r  =  — ae^'  [a  sin  x  —  cos  x)  4*  «^**  cos  ar  +  e«*  sm  x\ 

dx       a^  -\-  i   ^         ^  '  •* 

=  e**  sin  a;. 


CHAPTER    IV, 


SUCCESSIVE   DIFFERENTIATION. 


36    When  we  differentiate  a  function  u  =  Fx^  the  differential  co- 

efficient  —  will  usually  be  itself  a  function  of  x,  and  will  therefore 

admit  of  being  differentiated.  This  will  simply  be  equivalent  to 
examining  the  comparative  rates  of  increase  of  the   independent 

variable  x  and  the  variable  ratio  — •    This  differentiation  will  give 

rise  to  a  second  differential  coefficient,  which  may  also  be  a  function 
of  a;,  and  this,  in  its  turn,  being  differentiated  will  give  a  third  differ- 
ential  coefficient,  &c. 

37.  To  illustrate  this  subject,  let  w  ==  a;^  be  the  proposed  function. 
The  first  differential  coefficient, 

—  -  3a;2 
second  differential  coefficient. 


dx 


=  6a?, 


third   differential  coefficient, 


jdu 

a  — 

dx 

_     ~dx    _  ^ 

dx 

As  the  third  differential  coefficient  in  this  example  proves  con- 


46  DIFFERENTIAL  CALCULUS. 

stant,  the  fourth  and  all  succeeding  differential  C(/efficients  will  be 
equal  to  zero. 

38,  The  preceding  notation  of  successive  differential  coefficientft 
being  inconvenient,  it  is  replaced  by  the  following : 

du         . 

-,  dx  .  d^u 

£  or  — ; — J      we  write       — — ; 

dx  dx^  ' 

da 

e.^^  .      dx  .  oPm     „ 

^"^  d we  write      — — ?  &c., 

dx  dx^ 

~dx' 
the  symbols  d"^,  d^,  &c.,  indicating  the  repetition  of  the  process  of 
differentiation  twice,  thrice,  &;c.,  aaid  not  the  formation  of  a  power. 
On  the  contrary,  the  expressions  dx^,  dx^,  &;c.,  represent  powers 

d'^u 
of  dx.      The    second  differential  coefficient    -j-^    niay  be  obtained 

immediately  from  the  first  differential  coefficient   — ?    by    differen- 

dx 

thus  producing  -r-j 
and  then  dividing  the  result  by  dx. 

Now  since  the  law  according  to  which  the  independent  variable 
X  changes,  in  different  stages  of  its  magnitude,  is  entirely  arbitrary, 
we  adopt,  as  most  simple,  that  law  by  which  the  successive  incre- 
ments of  X  are  supposed  equal ;  that  is,  we  make  dx  constant. 
The  same  supposition  will  enable  us  to  derive  each  successive 
'  differential  coefficient  from  the  preceding  coefficient  by  a  similar 
process  of  differentiation  and  division. 

EXAMPLES. 

du  d'^u 

39.  1.  ti  =  x".         —  =  n.r«-i,     -—.  =  n(n  —  Ux*-^ 

dx  dx^  ^  ' 

~  =  n{n  -  l)(n  -  2)a;'-3^  ™  =  n{n  -  l){n  -•  2)(»  -  3)x— ^  &c. 


SUCCESSIVE   DIFFERENTIATION.  47 

This  operation  will  terminate  when  n  is  a  positive  integer ;  but 
if  n  be  a  negative  integer  or  a  fraction,  the  number  of  variable 
differential  coefficients  will  be  unlimited, 

dy       1      iPy  1       d^y       1.2 

d*y  1.2.3  ^,  ,  d^y       ^  1  . 2.  3  . . .  .(n  -  1) 

— —  = and  by  analogy     - —  =  dt -* 

dx^  a;*  dx""  x* 

the  upper  sign  will  apply  when  n  is  odd,  and  the  lower  when  n  is 
even. 

3.  tt  =  sin  X, 


du 

d^u 

dH 

dH 

—-  =  COS  X, 

-—-  =  •—  sm  X, 

-  cos  a;. 

dx 

dx^ 

dx^ 

dx* 

and  the  succeeding  differential  coefficients  will  recur  in  the  same 
order. 

4.  y  =  cos  X. 

dv  d^y  d^u        .  d*y 

and  the  coefficients  will  now  recur  in  the  same  order. 

5.  u  =  tan  X. 

^^  '.      dhi      ^      „  d^u       ^       -         o     ■   rt      ^      • 

— -  =  sec^ar,  -7—  =  2  sec^a; .  tan  x,  -r-r-  =  4  sec^ar  tan^a:  +  2  sec*ar,  &c» 
dx  dx^  dx^ 

Here  the  law  of  formation  of  the  successive  coefficients  is  not 
obvious. 

6.  «  =  a'. 

du  dhi  d?u 

—  =  a' .  log  a,      -^  =  a* .  log^a,     ^  =  «'  •  log^a,  <fcc, 

the  law  of  the  coefficients  being  very  evident. 


48  DIFFERENTIAL    CALCULtTS. 

7.  .  uz=ze*, 

di~^''    ■^~^''     "^"^'^ 
the  coefficients  being  all  equal. 

8.  u  —  sin(wa;). 

du  .     .      d^u  o  .    /     X    o 

-—  =  n  cosiwa:),    — —  =  —  n^&minx).  &c. 
dx  ^  dx^  ^     ' 

The  formation  of  successive  differential  coefficients  will  be  found 
extremely  useful  in  the  expansion  of  functions  by  the  methods 
which  will  be  explained  in  the  chapters  immediately  succeeding. 


I /in  U  A  H  Y  \ 
UN  I  ^■Kl{sn'^'  op 
i   CAtJKoi:NlA. 


V... 


CHAPTER  V. 


MACLAURIN  S    THEOREM. 


40.  The  theory  of  Maclaurin  is  a  very  general  and  useful  formula 
for  the  development  or  expansion  of  a  function  of  a  single  variable, 
in  a  series  involving  the  positive  ascending  powers  of  that  variable, 
when  such  development  is  possible. 

41.  Prop.  If  y  =  Fx,  where  Fx  denotes  such  a  function  of  a;  as 
can  be  expanded  in  a  series  containing  the  positive  ascending  powers 
of  a:,  then  will  the  form  of  the  development  be  the  following : 

in  which  the  parentheses  are  used  to  denote  the  particular  values  of 

dtj    d'^ii 
the  quantities  y,  ~,  --7-^,  &c.,  enclosed  therein,  when   x  is  taken 

equal  to  zero. 

Proof.  By  hypothesis,  y  can  be  expressed  in  the  form 

y  =  A-{-Bx+Cx'^  +  Dx^-\-  Ex*  +  &c.,         (1), 

in  which  A,  B,  (7,  &c.,  are  unknown  constants. 

.  •.  ^  =  B  +  2Cx-^-  SBx"  +  4Fx^  +  &c. 
ax 

^=2C+2.SDx  +  S.4Ez^  +  &(i, 
dx^ 

■^  =  2 . 3  i>  +  2 .  3 . 4 JKr -f  &c. 

dx^  a 

^=3  2.3.4^+ &c. 

&c.         dec. 
4 


50  DIFFERENTIAL   CALCULUS. 

Now  making   x  —  0   in  each  of  these  expressions,  we  obtain 

.-..=,„,  .=(*),  <-=i(S)=ji,(S). 

These  values,  being  substituted  in  (1),  reduce  it  to  the  form 

which  agrees  with  the  enunciation. 

This  formula,  called  Maclaurin's  Theorem,  may  be  written  thus 

or  again,  if  we  represent  the  1st,  2d,  3d,  &c.,  differential  coeffi- 
cients, which  are  functions  of  x,  by  FiX,  F^^  F^x^  &c.,  the  formula 
may  be  written 


1   '      ^    1.2         ^   1.2.3 


+  ^*' 1:2:3^+^"         W- 


EXAMPLES. 


42.  1.  To  expand       i^^j)  =  (a  +  a?)". 


MAclapkin's  theorem.  51 

Here  g  =  «{«  +  ^)"-\      "^  =  «(»  -!)(«  +  =')-"> 
-g  =  »(«-!)(» -2)  (a +  »)-', 


-T^  =  n(n  —  1)  (»  —  2)  (»  —  3)  (a  +  a;)"-*,  &o.,  &c. 
Hence,  when  a;  =  0. 

(S)  =  '^(^  ^  1)  (»*  -  2)  (^  -  3)a«~*,  &c.,  &c. 

And,  therefore,  by  substitution  in  Maclaurin's  formula, 

y  =  (a  -f-  a;)"  =  a«  +  na»-iar  +  ^\  ~    ^  a^'-^x^ 

n(n-\){n-2) 
+  1.2.3  ""     "^ 

Thus  we  have  a  simple  proof  of  the  binomial  theorem,  applicable 
to  all  values  of  the  exponent,  whether  positive  or  negative,  integral 
or  fractional,  real  or  imaginary. 

2.  To  develop  y  =  log  (1  +  x), 

the  modulus  of  the  system  being  M, 

dy         M        d^y  M  d^y         1.2if 


dx       \  -\-x      dx^  (1  +  xY      dx^       (1  4-  x) 

d^y  _        1 .  2 .  3  Jf 
d^-        (1  +  xY'    ^''' 


52  DIFFERENTIAL   CALOULUS. 

.  • .  when  X  =  0,     {y)  =  log  1=0, 

idy\      M  l<Py\         M  l(Pq\      \  .2M  ld^y\  1.2.3.lf   , 

i^;^  r  wr^T  wr—i-'  w) = — ^1 — '  ^^ 

And  by  substituting  these  values  in  Maclaurin's  f  rmula,  we  have  ^- 
y  z=z  log  (1  +  a:)  =  M{x  —  -^  x^  -\-  -  x^  —  -  x^  -\-  &c.) 

/w  o  4 

which  is  the  fundamental  theorem  used  in  the  computation  of  loga 
rithms,  and  is,  indeed,  that  which  was  employed  in  deducing  the 
rule  for  differentiating  logarithms. 

3.  To  expand  y  =  sin  x. 

Here  Fx  z=is\nx 

. ' .  F-^x  =z  cos  Xj     F^x  =  —  sin  ar,     F^x  =  —  cos  x,     F^x  =  sin  x, 

and  the  succeeding  coefficients  recur  in  the  same  order. 

.'.  FQ  =  sin  0  =  0,    F^O  =r  cos  0  =  1,    F^Q  =  0,    F^O  =  -  1, 

F,0  rr  0,    F^O  =  1,    &c. 

.  • .  by  substitution  in   (4)   the  third  form  of  Maclaurin's  theorem, 

we  have 

x^        .  x^  x"^ 

&mx  =z  X f-  (Sec. 

1.2.3  ^1.2.3.4.5       1.2.3.4.5.6.7^ 

This  series  converges  very  rapidly  when  x  is  small. 

\/     4.  To  expand  y  =  cos  x. 

Fx  =  cos  X,  F^x  ~  —  sin  x,  F2X  —  —  cos  ar,  F^x  =  sin  x,  F^x  =  cos  x, 

and  the  succeeding  coefficients  recur  in  the  same  order, 

.■.F0  =  1,  F^O  =  0,  F^O  =  -  1,  F^O  =  0,  F^O  =  1,  F,0  =  0,  &c. 

.  • .  cos  ar  =  1  —  - — -  + ^  &c. 

1.2^1.2.3.4       1.2.3.4.5.0^ 


tmaclaurin's  theorem.  58 

5.  To  develop  y  =  a*. 

Employing  Naperian   logarithms,  we  have 

Fx  z=  a',    F^x  —  a*,  log  a,    F2X  =  a* .  log^a,     F^^x  =  a*.  log%,  &c 

r,FO=:l,FyQ  =  log  a,  F2O  z=  log^a,  7^30  =  log^a,  F^O  =  log^a,  &«. 

X  x''"  x^ 

.-.  o'  =  1  +  logay  +  log^a  j-^  4-  log'a-j-^-g 

This  is  called  the  exponential  theorem. 

Cor.  Jf  a  =  c  the  Naperian  base,  then  log  a  =  loge  =  1, 

••-'  =  » +  i  +  o+r:2:3  +  r72:3r4  +  *'^' 

and  if    a;  =  1     also, 

a  formula  for  the  Naperian  base. 

Cor.  If  a:  =  1,  but  a  not  equal  e,  then 

a  r=  1  4-  log  a  +  Y"^  log^a  +  ^    ^    ^  V'g^a  +  772^74  l^g*«  +  *^«- 

a  formula  for  a  number  in  terms  of  its  Naperian  logarithm. 

Prop.  To   express   the   sine   and  cosine  of  an   arc  in   terms  of 
imaginary  exponentials. 

In  the  series  giving  the  value  of  e*,  put  successively 

z^—  1,       and      —  z^—  1  ^^^  ^• 


1  1.2       1.2.3        1.2.3.4 

+  1.2    3.4.5-^^- 


54 

and 


DIFFERENTIAL   CALCULUS. 


1  1.2^  1.2.3 


+ 


1.2.3.4       1.2.3.4.5 


—  (fee. 


But  the  first  series  within  the  [  ]  is  the  development  of  cos  2,  and 
the  second  that  of  sin  z. 


.  • .  cos  2  = 


g«v^^+  e-V-i 


(^), 


These  singular  formulae,  discovered  by  Euler,  are  very  useful  in 
the  higher  branches  of  analysis,  especially  in  the  development  of 
functions. 

Cor.     ]f  we  divide  {JB)  by  (.4),  there  will  result 


tang  = 


.V=r 


g2zv'=r_  1 


-  l[e^v^i  ^  e-^V^        /ZrT[e2zv^=r_^  1] 


(C). 


Cor.  If  we  make  z  =  xy—l  in  (^),  (^),  and  (C),  we  can 
express  the  sine,  cosine,  and  tangent  of  an  imaginary  arc  in  terms 
of  real  exponentials  ;  thus  : 

8in  (^^/^ly  =  .^^. . .  (D),  cos  (^/3T)  =  fll+i!. . .  (r) 


2V 
tan  (a;-/—  1) 


e-2x  _  1 


1  -e^' 


"^^(e-^'  -f  1)        V-  HI  +  «^') 


55 
Cor.  If  we  square  [A)  and  {B)  and  add,  there  will  result 

cos^z  -f  sin^z  = =  1. 

And  similarly     s,m^(x^—  1)  4*  cos^^^r^—  1)  =  1; 

two  results  obviously  correct. 

43.  The  applications  of  Maclaurin's  theorem  are  often  much 
restricted  by  the  great  labor  necessary  in  forming  the  successive 
differential  coefficients.  This  may  sometimes  be  avoided  by  ex- 
panding the  first  differential  coefficient  by  some  of  the  algebraic 
processes.     For  example, 

To  expand  u  =  tan~^;c. 

,_  du  1 

which  gives  by  actual  division,  the  quotient 

1  —  x^  -^  x^  —  x^  -\-  x^  —  &c. 
.     .  • .  Fx  =.  tan-i;r, 

FyX  z=:   \   —  X?  -{-  X^  —  X^  -{-  X^  —  &C. 

F^x  =z  —  2x  -f-  4x^  —  Qx^  +  Sx"^  —  &c. 

i^3a;  ==  -  2  +  3.  4a:2  -.  5  .  6a:*  +  7  .  ^x^  -  &c. 

/;a;  =  2.3.42:  -  4.5. 62;3  -f  Q.l  .%x^  -  &;c. 

i?;^:  =  2  .  3 . 4  -  3  .  4 .  5 .  62;2  +  5 .  6 .  7 .  8a;*  ~  (fee. 

i<;a:  =  -2.3.4.5.6a;  +  4.  5.6.7.  8a;3  -&c. 

F^x=  -2.3.4.5.6  +  3.4.5.6.7.8a;2-&c. 
'  i^ga;  =  2.3.4.5.6.7.8a;-&c. 

&c.,  &c. 

.-.  i?^0  =  tan-iO  =0,     i^iO  =  l,     F^O  =  0,     Fjd  =  -  1 .2, 
Ffi  =  0     i^sO  =  1.2.3.4,     FqO  =  0, 
F^0=-  1.2.3.4.5.6,     i^80=0,&c. 


56  DIFFKRENTIAL    CALCULUS. 

Therefore,  by  substitution  in  Maclaurin's  formula, 

Fx  =  tm-^x  =  X  —  -x^  +  - x^  —  -x'  -t  -x^  —  &c. 

o  o  i  ^ 

If,  in  this  formula,  we  make    u  =  -'^^  —  arc  of  45°, 
then  ir=tan45°  =  l. 

•  •  r=o-3+5-7  +  ^^-)' 

and  'T'  =  4(1  -  -  -h  ^  —  i^  +  &c.); 

a  formula  for  determining  the  ratio  of  the  diameter  to  the  cir- 
cumference of  a  circle. 

This  series  converges  so  very  slowly,  that  even  a  tolerably 
accurate  approximation  to  the  value  of  -tt  cannot  be  deduced  from 
it,  without  employing  a  great  number  of  terms. 

44.  Frop.  To  deduce  Euler's  more  convergent  series  for  the 
ratio  of  the  diameter  to  the  circumference. 

If  in  the  trigonometrical  formula 

,  , ,  tan  a  -\-  tan  b 

tan  (a  -f-  6)  = ,i 

^  ^       1  —  tan  u  .  tan  b 

we  put         a  +  ^  =  T  *'         ^^^^         ^"  (^  -f  ^)  =  1> 
.  • .     1  •—  tan  a .  tan  b  =  tan  a  -{-  tan  6 ; 

1  -.    -.  7        1  —  t^"  ^ 

whence  we  deduce  tan  o  =  - — ; • 

1  +  tan  a 

And,  therefore,  if  any  value  be  assigned  to  tan  a,  that  of  tan  * 
can  be  determined. 

Let  tan  a  =  - ,        then  tan  b  = =  -• 

1  il    .  ,1 

.  • .  T  *  =  tan-1  -  4-  tan-i-. 
4  5«  o 


MAlDL^URiN's  THEOREM.  57 

B..      „-,i  =  l_>(i)V!(!)-_>(!)V^  . 

and  tan-l.l_JQ%jg_jg+&c. 

•'•4^  =  2-3:25  +  5:2^' -tT'^-^"- 

■+3       3  .  33  +  5  .  3^       7.3^  + 

By  taking  six  terms  in  the  first  set,  and  four  in  the  second,  and  mul- 
tiplying by  4,  we  get  the  common  approximation, 

-^r^  3. 1416. 

Cor.  We  might  extend  this  method,  obtaining  series  still  more 
convergent.      For  if  we  take  four  arcs  Cj,  Cg,  03,  and  c^,  such  that 

Cj  4-  ^2  ==  tan-^  -  and  Cg  +  c^  =  tan"*^  -.    Then  Ci+  <^2+  ^3+  ^4=  ""*> 

and  if  we  assume  the  values  of  tan  Tj  and  tan  Cg,  those  of  tan  Cg  and 
tanc^  can  be  determined.  Moreover,  the  values  of  tan  c^,  tan  Cj. 
tan  C3,  and  tan  c^,  can  all  be  repdered  less  than  i,  and  therefore  th6 
series  for  determining  \  ir  will  be  more  convergent. 

45.  Prop.  To  obtain  more  convergent  series  for  the  value  of  if, 

2  tan  a 


If  in  the  formula  tan  2a  = 


1  —  tan2tt' 


1 

we  put  tana  =  r,     then 

o 

^     o                 5        5 
tan  2a  = =  — -» 

25 

2xA 

2  tan  2a  12        120 

and  •  •  •  jBH  4a  = = =  — • 

•   '   -^-^         I  _  tan2  2a       .  _  2b^       119 

144 


58  DIFFERENTIAL    CALCULUS. 

Now  this  result  is  very  little  greater  than  unity,  and  therefore  4a 
must  be  slightly  greater  than  45°. 

Put 


Then 


4a  —  -  <:r  = 
4 

:  Z 

a  very 

small  arc. 

tang  = 

tan  4a  ■-  tan  -  •»' 
4 

tan  — ' 
4 

1  +  tan  4a. 

r 

120       , 

119       ' 

1 

-          120- 
^119 

'239 

• 

• .  -  *  =  4  tan-i  -  ■ 
4                       5 

-'^■^-'239 

4- 

_     1     +     1 
3.53   *  5.55      7 

1  +  1 

—  &c 

') 

.5'   '   9.5« 

V239      3.2393   '  5.2395 

—  &c 

) 

By  taking  three  terms  in  the  first  line  and  one  in  the  second,  we  get 
the  common  approximation  <:r=:3.1416;  and  by  taking  eight 
terms  of  the  first  line  and  three  of  the  second,  We  get 

-;r  =  3.141592653589793. 
46.  1.  To  expand  u  =  sin-^ic. 

J^x    =  sin~i  X. 

„     1.2   ,  1.3.4  3  ,  1.3.5.6,  ,  . 
^'^  =  — 2*  + 1:2:2-^^  +  r273-:25^'  +  ^''■ 

_         P.2,  1.32.4   ,  ,   1.3. .52. 6    ,   ,    . 
'^'^  =^  -IT2  +  17272^^'  +  1727372-3^  +  ^- 


maclaurin's  theorem.  ,      59 

1.2.32.4     ,  1.3.4.52.6  „  ,    . 

-^^^^17272^^+     1.2.3.23    ^'  +  ^'' 

*  1.2.22     -^     1.2.3.23     -^   ^^^• 

_,  1.2.32.4.52.6      ^. 

^^'=        1.2.3.23       ^  +  ^^' 

12.2.32.4.52.6   ,    , 
^^"^        1.2.3.23        +'^^' 

•  /;= 0,  i^iO  =  1,  F^o  =  0,  F^o  =  12,  i<;o  =  0,  f.o  =  p. 32, 

i^fiO^O,  F^0  =  12.32.52,  &c. 

.     ,  12.a;3  12.32.a;5 


1.2.3^1.2.3.4.5 

12.32.52.0:' 


1.2.3.4.5.6.7 


-i-  &C. 


CHAPTER    VI. 


47.  Taylor's  Theorem  is  a  general  formula  for  the  development 
of  a  function  of  the  algebraic  sum  of  two  variables. 

Proi[t.  \t'  1/  =  Fx,  and  if  x  be  supposed  to  receive  an  increment 
h,  converting  y  into  y^  =  F{x  +  ^)  ;  th*^n  will 

dy   h   ,    d^y      h'^     '    d^y         h^  d^y  h* 

^,     ,    ,,        ^     ,   dFx    h   ^  d^Fx     li' 

d^Fx        h^  d^Fx         A^ 

dx^  '1.2.3        dx*   *  1.2.8.4"^     ^' 

To  prove  the  truth  of  this  formula,  we  first  establish  the  following 
principle  : 

If  in  the  expression  y^  =  F(x  -\-  h)  we  suppose  first  that  x  is 
variable  and  h  constant,  and  then  suppose  h  variable  and  x  constant, 
the   first  differential   coeflicient  will   be   the   same  in  both  cases; 

&  =  '! 

This  is  almost  self-evident,  for  when  a  given  increment  is  assigned 
to  X,  or  to  h  the  same  increment  must  be  imparted  to  ar  +  A,  and 
therefore  F{x  -|-  h)  =z  y^  will  undergo  the  same  change  in  the  one 
case  as  in  the  other.  Hence  the  ratio  of  the  corresponding  change* 
of  X  and  yj  is  equal  to  the  latio  of  the  changes  in  h  and  y^  This 
is  true  whatever  may  be  the  magnitudes  of  the  increments  im 


Taylor's  theorem.  61 

parted  to  x  or  A,  provided  that  magnitude  be  the  same  in  both 
cases.  But  when  we  suppose  these  increments  indefinitely  small,  it 
is  no  longer  necessary  to  consider  them  equal.     For  since  the  ratio 

-^  does  not  contain  dx,  it  will  have  the  same  value  whether  dx. 
dx 

and  dh  be  supposed  equal  or  unequal. 


.    dy^       #1 
'   '  dx  ~  dh 

dx      ~ 

dx               dh 

d^y^       d^y^ 
dx^        dh-'*' 

d'^y^      d:^y^ 
dx''  -  dh'' 

Similarly, 

And  generally, 

Now  assume 

yi  =  F{x  +  A)  =  i^ar  +  ^A  +  Bh''  +  Ch^  +  JDh^  +  &c.       (1), 

that  being  the  general  form  in  which  F(x  4-  h)  can  be  developed, 
as  shown  in  Art.  4.  The  coefficients  A^  B,  C,  i>,  &c.,  are  func- 
tions of  x,  but  are  independent  of  h. 

If  we  differentiate  (1)  first  with  respect  to  h  and  then  with 
respect  to  ar,  and  place  the  resulting  differential  coefficients  equal, 
we  shall  obtain 

A  +  2Bh  4-  3  Ch^  +  4i)A3  +  &c. 

dFxdA  dB^^dC^^^^ 

which  equation  being  true  for  all  values  of  A,  it  follows,  by  the 
principle  of  indeterminate  coefficients,  that  the  coefficients  of  the  like 
powers  of  h,  in  the  two  members  of  the  equation,  must  be  sepa- 
rately equal. 

dx    .  dx  dx  dx^ 


62  DIFFERENTIAL   CALCULUS. 

A  -—        p  _  1  ^^  _     1       d-'Fx 
•*•  dx'  ~2dx~  1.2'  dx^  ' 

1    c?5  _       1         d^Fx      j._l^_ ]__  d*Fx 

3  '  ^  ~  1.2.3  '  ~d^'  ~'i"d~x~  1.2.3.1'  dx*  '    '^ 

Hence,  by  substitution  in     (1), 

^,     .    ,,       ^        </i^;c  ^       d'^Fx     h?        d'Fx        h? 

■       "^-^-172:3:4+  ^^^ 

dy   h       d'^y      h?        d^y         h^ 

or,  y^::3y4._. _  +  _.__+_. _____ 

^  ^;r*   1.2.3.4^ 

]f  we  denote  the  successive  clifTerential  coefficients  by  F-^x,  F^ 
F^x,  F^x^  &;c.,  the  series  may  be  written 

J?  ^2  ^3 

Cor.  The  formula  of  Maclaurin  may  be  readily   deduced  from 
that  of  Taylor;    for  if  we  make  x  =  0  in  (2),  there  will  result 

Fh  =  FO  +  Ffi  -  +  /iO  —  +  F,0  —-- 
which  is  Ma'claurin's  theorem. 

EXAMPLES. 

48.  1.  To  expand    sin  (x -\- h),   in  terms  of  the  powers  of  the 

aic^. 

F{x-{-  h)  =zsm{x  +  h), 


Taylor's  theorem.  63 

.  • .    Fx  z=z  sin  x^       F^x  =  cos  a:,       F^x  =  —  sin  x, 

F^x  =  —  cos  X,       F^x  =z  sin  x^  &c. 

.  * .  By  substitution  in  Taylor's  formula 

h  h^  h^ 

liii  (x  -\-  h)  r=  sin  a;  +  cos  a;  -  —  sin  x- — -  —  cos  x  - — -— -  -f-  <tec. 

+  cos  ^  (A  -  j-^  + -j-^-^  -  &c ) 

=  sin  a? .  cos  h  +  cos  a; .  sin  A,   a  well  known  formula. 

2.  To    expand     cos  [x  -f-  A),    in   terms    of  the    powers    of  the 

arc  h, 

F  (x  +  h)  =  cos  (x  +  h), 

.' .    Fx=  cos  X,     F^x  =  —  sin  x,     F^x  =  —  cos  ar, 
F^x  =  sin  a;,     /"^a;  =  cos  x,  &;c. 
.  • .  By  substitution  in  Taylor's  Theorem  we  have 

cos  («  -j-  A)  =  cos  a:  —  sm  ic  -  —  cos  a:  - — -  +  sm  x 

=  cos.r(l-  — +  j-^^-^^-&c.,) 

7^3  ^5 

=  cos  X  .  COS  A  —  sin  a; .  sin  A, . .  .  a  well  known  formula. 

3.  To   expand     log  (a; -f  A),    where  M  is   the   modulus  of  the 
system. 

i^aj  =  logar,     ^1^  =  -,     i^ga:  =  -  — , 

if^  =  — -—,    jb^x  = — ,  &c., 


64  DIFFERENTIAL   CALCULUS. 

...  log(^  +  A)=log.:  +  if^--  — +  3^-— +&C.) 
4.  To  expand  Wj  =  tan-i(a;  +  ^), 

u  =  tan-'^x  =  Fx,    F.x  =  - — ; — -  =  — —  =  cos^u, 

F^  =  —  2sin  u .  cos  u—  =  —  sin  2w  .  cos%. 

F^x  =  (—  2cos  2w .  cos'^u  -}-  2sin  2m  .  cos  m  .  sin  m)  — 

=  —  2cos  u  .  cos  Su  -y-  =  —•  2cos  3m  .  cos^w. 
ax 

FaX  =  2.3  (sin  3m  .  cos^m  +  cos  3m  .  cos^m  .  sin  u)  — 
*  ^  ^  dx 

=  2 .  3cos2m  .  sin  4M-r-  =  2  .  3  .  sin  4m  .  cos*m, 
ax 

&c.,  &c. 

k  h!^ 

•  • .  tan~i  (x  -\-  h)  =2  u-^  z=i  u  -{■  cos^m sin  2m  .  cos^m  ~ 

7^3  7^4  7^5 

—  COS  Zii .  cos^w  — -  +  sin  4m  .  cos%  — -  -f  cos  5m  .  cos^m  — &c. 

3  4  5 

6.  To  expand  u  =  tan(.T  -f-  h). 

Fx  =  tan  X,      F^x  =  seo'^x,      F^x  =  2  sec^a: .  tan  a;, 

i^go;  =  2  sec^o;  (1  +  3  tan2a:).      &c.,     &c.  N^ 

.  • .  tan  {x  +  h)  =  tan  x  +  sec^a;  -  +  2  sec^a; .  tan  x  - — - 

1  1  .  /i 

7^3 

+  2  sec^a;  (1  +  3  tan2a;)  r—,r-r,  +  &c. 
1  .  2  .  o 

Prop.  Having   given      u  =  Fy,      and      y  =  dpx^      to   form   the 

differential  coefficient  —  of  m  with  respect  to  a:,  without  eliminating 

y  "between  the  equations,  in  which  the  characters  F  and  9,  denote 
any  functions  whatever. 


TAYLOR'S  THEOREM.  65 

Let  X  take  an  increment  h  converting  y  into  y^^  —  ^{x  -\-  A). 
Then  if  k  denote  the  increment  received  by  y.  we  shall  have,  by- 
Taylor's  theorem, 

Also  when  y  takes  the  increment  k^  it  imparts  to  w  =  Fy.  an 
increment 

„,         ,.        _,        du   k      d'^u     P        d?u       P 

or  by  substituting  for  k  its  value,     (1).  « 

duVdy  h       d?-y     hP'         d^y        h^        ,    ^      1 

1  . 2   dy^  Ldx    1       ax^    ]  .  2  J 

Dividing  both  members  by  A,  and  then  passing  to  the  limit  by 

-..-.,.,  21-.  —  u         du 

making  h  =  0,  in  which  case     — ^-r —  =  —     we  get 

ii  ax 

Thus  it  appears  that  the  differential  coefficient  of  u  with  respect 
to  X,  is  found  by  differentiating  u  as  though  y  were  the  inde- 
pendent variable,  then  differentiating  y  as  th(jugh  x  were  the 
independent  variable,  and  finally,  multiplying  the  first  of  the  co- 
efficients so  found  by  the  second. 

49.  It  might  perhaps  seem  at  first  view  that  the  equation  (2)  is 
necessarily  and  identically  true,  and  therefore  that  the  precedmg 
investigation  is  unnecessary.     But  it  must  be  borne  in  mind  that  the 

dy  which  appears  in  the  coefficient   -j-    and  which   represents   (ho 

increment  given  to  y  by  assigning  an  arbitrary  small  increni-  i'   dx 
to  the  variable  x,  is  not  necessarily  the  same  as  dy  which  ap{)cairf  in 
5 


60  DIFFERENTIAL   CALCULUS. 

du 

— ,  since  this  latter  increment  of  y  is  arbitrary   (though  likewi"' 

small). 

^    ^      du 

1.  u  =  a^j  y  =  0*,     to  nnd      — • 

du  .  dy       ,    .      . 

Here  —  =  ay.]oga,     -—^b'logb. 

du       du    dy  ,      ,  ,       ,  lx    ,      ,  ,7 

.♦.■—  =  -; r-  —  av .  b^  Aos  a .  \os  b .  =  a^   .  6*  .  log  a  .  log  6. 

dx       dy    dx  &  5  fe  & 

2.  w  =  log  y.     y  z=i  log  a;. 

c/m       1      c?y       1  du       1     1  _       1 

dy  ~  y^     dx~  x'      '    '  dx~  y    x~  xlogx 

50.  Taylor's  Theorem  may  be  employed  in  approximating  to  the 
roots  of  numerical  equations. 

Let  Fx  =  0  be  the  given  equation,  and  a  an  approximate  vahie 
of  one  of  its  roots  found  by  trial ;  then  we  may  put  x  =  a  -{-  h,  in 
which  A  is  a  small  fraction  whose  higher  powers  will  be  small  in 
comparison  with  A,  and  may  therefore  be  nejjlected  without  great 

error.  .  But 

h  K^  h^ 

Fx  =  F(a  +  h)  =  Fa  +  F,a.j  +  ^2«  y^  +  ^3«  XT^^g  +  <^c.  =  0. 

.  • .  By  neglecting  the  terms  involving  ^2?  ^^?  <^c.,  we  get 

Fa-\-  F.a~  =  0       and       .'.h  =  —  -jr— 
^    1  F^a 

Adding  this  approximate  value  of  h  to  a,  we  have 

Fa  . 

X  =z  a  — FT-    nearly. 
F^a  •' 

Call  this  value  a^     and  put     x  =  a^  -\'  h^ 

Then  by  similar  reasonmg  we  shall  find 

A,  = ~-,     and     x  =  a^ z~-  =  aj'  ^  nearer  approximation, 

and  the  same  process  may  be  repeated  if  necessary. 


67 

51.  Find  tlie  positive  root  of  the  equation 

x^  ~  12a:2  -f  122;  -  3  =  e 
to  three  places  of  decimals  inclusive. 
Here  we  find  by  trial  that 

a;  >  2  .  6        and        ar  <  3. 
Put  cf  =  2 . 8. 

.'.  Faz=za^  -  \2a?  -|-  12a  —  3 

=  (2.8)*-  12(2.8)2+  12(2.8)-3=  -2.0144 

dWn 

F^az=z  ~-  =4a3-24a+  12  =  4(2.8)3-24(2.8)4-12  =  32.608, 

—  2.0144 

.«.  A  =  -  — -^-— -  =  0.062.nearly.  .'.x=:a  +  ^  =  2.862  nearly 

To  test  the  accuracy  of  this  approximation,  put 
«!  =  2 .  802         and         x  --a^-^  h^. 
Fa^  =  (2 .  862)*-  12(2 .  862)2+  12(2 .  862)  -3=0 .  144674  nearly 
F^a.^  =  4(2 .  862)3  _  24(2 .  862)  -f  12  =  37 .  083072  nearly. 
0 . 144674 

.•.a;  =  ai  + Ai  =  2.862  -0.003901  =2.858099  =  2.858 

to  three  places  of  decimals. 

If  the  process  were  repeated  it  would  be  found  that 
a:  =  2 .  85808  ; 

so  that  the  second  approximation  is  true  to  foui  places  of  decimals, 
and  the  fifth  place  is  slightly  erroneous. 
2.  Given  ^*  =  100 

to  find  the  value  of  x  to  the  place  of  hundredths. 
Passing  to  the  common  logarithms,  we  have 

X  log  X  =  log  100  =  2.      .  • .  a:  log  a;  —  2  =  0. 

Also  ^  >  3,       and       a;  <  4. 


68  DIFFEKENTIAL    CALCULUS. 

IRut  a  ==  3 . 5,       and       x  =  a  -{-  k. 

,'.  Fa  z=z  a  log  a  —  2,       F^a  =  — -^  =  log  a  -f  -^, 
where  M  =  modulus  of  the  common  system  =  .  43429448 
Fa   =3.51og(3.5)  -2=  .544068  X  3.5-2=  —0.01)5762 
F,a  =  .  544068  -f  •  434294  =  0 .  978362. 
^       .0957<)2 

•••*=r97s3oi  =  •«««• 

.•.a?=:  3.5 -f.  098  =  3.598      or      a:  =  3  .  60  nearly. 

We  shall  now  apply  Taylor's  Theorem  in  deducing  rules  for  the 
rtxpansion  and  differentiation  of  functions  of  more  complicated 
forms. 

52.  Prop.  To  establish  a  general  rule  for  differentiating  any 
function  of  two  quantities  p  and  q,  which  quantities  are  themselves 
functions  of  the  single  independent  variable  x. 

Let  u  —  F(p,  q),  where  p  =/x,  and  q  =fjX,  the  characters 
F,f,  and /^,  denoting  any  function  whatever,  and  let  x  take  the 
increment  h,  converting  p  in  p  +  k  =  p^,  q  into  q  -{•  I  =  qi,  and 
u  into  Uy 

Then  n,  =  F(p  -^  k,  q -^  I)  =  F(p  -f  k,  q,), 

which  may  be  developed  by  Taylor's  Theorem  as  a  function  of 
p  +  A:,  observing  that  q-^,  which  does  not  contain  k,  will  appear  in 
the  development  as  would  a  constant : 

.-.«.  =  ^-C/- +  *,?,)=  ^U  ?.)  + ^,^ -Y 

+  f£^.f^^^,.    (1). 

dp''  1.2  ^  ' 

But  F  (p,  ^i)  =  P(Pi  q  +  /),  which  developed  as  a  function  of 
q  \-  I,  gives 


TAYLOR'S  THEOREM.  69 

And  similarly  the  coefficient  of  k  in  the  second  term  of       (1). 

dF(n,q,)  ^  dF(p,q^-l) 
dp  dp 

may  also  be  developed  as  a  function  of  5-  -f  ^,  and  will  give 

dF{p.q-\-l)  _  dF(p,q)         d  rdF{p,  q)-\  I 

dp  -         dp  ^  dqL        dp        Jl"^*^"^ 

And  in  like  manner 

<PF{p.q,)   _  d^Fjp,  q  +  l)  _  d^Fjp^q)         d_  frf^^l 

dpP'  ~  dp''  -         dp''         '^  dqL       dp'       J"^^"^ 

.'.By  substitution  in     (1). 

H-  terms  involving  P,  kl^  P^  P,  &a 
o  ^  T  dp    h       d'p     h?" 

duTdq    h  ^d'q      7.2  1 

^  c^yLt/a:    1^6^x2    1.2^        'J 

duVdp    h       d''p     W  1  ,    * 

Now  dividing  by    A,  and  then  passing  to  the  limit,  by  making 

h  =:  0,  m  winch  case  — ^— —  =  -^,  we  obtam 
h  dx 

du      du    dq      du   dp  .  . 

dx  ~  dq    dx      dp    dx  ^  '* 

.■.^~dx  =  du  =  '!^.'^dx  +  -.'^^4x.        (3). 
dx  dq    dx  dp    dx  ^  ' 


70  DIFFERENTIAL   CALCULUS. 

Thus  it  appears  that  we  must  differentiate  u  with  respect  to 
each  function,  as  though  the  other  functions  were  constant,  and 
add  the  results. 

53.  It  is  very  important  that  the  precise  signification  of  the 
notation  here  employed  should  be  distinctly  understood.  By  an 
attentive  consideration  of  the  manner  in  which  the  several  expres- 
sions employed  in  the  formulae  (2)   and   (8)  arise,  it  will   appear 

that  the  expression  ^in   (2),  represents  the  ratio  of  the  change 

in  X  to  the  entire  change  in  w,  which  latter  is  produced  partly  by 
the   change   imparted   to   ^,  and   partly  by  that   imparted   to    q\ 

that  the  expression —  represents  the  ratio  of  the  change  in  x 

to  that  part  of  the  change  in  u  which   is   communicated  through 

q :  and  that  —  •  ~    represents   the   ratio  of  the  change   in   x  to 

that  part  of  the  change  in  u,  which  is  communicated  through  p. 

du    dq  du 

dq    dx '  dx* 

or  to  suppose  that  the  first  of  these  expressions  can  be  brought 
to  the  form  of  the  second  by  the  ordinary  process  of  algebraic 
reduction.      This  will  appear  evident,  when  it  is  recollected  that 

the  du  which  appears  in  —  refers  to  the  total  change  in  ?/,  while 
ax 

the  da  which  occurs   in f-.  refers    only    to    so    much    of  the 

dq    dx'  -^ 

change   in   u,  as    is  communicated  through  q.     Similarly,  —  •  -p, 

dti 
must  not  be  confounded  with  -7-,  for  a  like  reason. 

dx 

54.  To  differentiate  u  —  F{p^  q,  r,  s,  &c.)  when  p,  q,  r,  5,  &c. 
are  functions  of  the  same  variable  x. 

By  attributing  to  x  an  increment  A,  and  reasoning  as  in  the  last 
proposition,  we  readily  prove  that 


We  nmst  be  careful,  therefore,  not  to  confound  j^-*  33-,  with 


Taylor's  theorem.  71 

fdii.   dp      du    dq      du    dr      du    ds  "]A 

4-  terms  in  A^,  A^,  &c. 

T»^rt<posmg  M,  dividing  by  A,  and  then  passing  to  the  limit,  we  hav« 

du      du   dp       du   dq       du    dr   ,    da    ds    ,     . 

—  = ±1-| 1-j 1 . 1-  &c. 

dx      dp    dx       dq    dx       dr   dx       ds    dx 

du  ,  ,         du   dp     .      ,   du   dq     ^         du    dr     . 

.-,  -—dx  =du  =  - f- ' dx  +  -J-  '-y- '  dx  -{-  -J- ' -J- ' dx 

dx  dp   dx  dq  dx  dr   ax 

du    ds 

'^'Ts'di'^'^'^^''' 

that  is,  we  must  differentiate  u  with  respect  to  each  of  the  functions, 
as  if  the  other  functions  were  constant,  and  add  the  results. 
55.  Prop.  To  differentiate  u  =  F(p,  a;),  where  p  —  fx. 
Here  u  is  directly  a  function  of  x,  and  also  indirectly  a  function  of 
X  through  p. 

Now  if  in  the  equation      u  =  F{p,q),  which  gives 

du       du    dp       du    dq 
dx  ~  dp   dx       dq   dx 

we  put  q  =  Xy  there  will  result 

^     du      du   dp       du    dx 
"~     ^^'    "^  dx~  dp    dx       dx    dx 

du      du    dp      du       ,  .         .  dx 

dx  ~~  dp    dx      dx       ^  '^  dx~    ' 

The  formula  (1)  is  that  required,  but  we  must  distinguish  care- 
fully between  the  differential  coefficient  —  in  the  first  member,  and 

the  similar  expression  in  the  second.  The  latter,  called  the  partial 
differential  coefficient  of  u  with  respect  to  a:,  refers  only  to  that  part 
of  the  change  in  u  which  results  directly  from  a  change  in  x,  while  p 
is  supposed  to  remain  constant ;  and  the  former,  called  the  total  dif- 


72  DIFFERENTIAL   CALCULUS. 

fereiitial  coefficient  of  u  with  rospect  to  x^  refers  to  the  entire  changt 
in  u.  which  is  partly  th^  direct  result  of  a  change  in  x,  and  partly 
an  indirect  effect  produced  through  jo. 

To  distinguish  the  total  from  the  partial  differential  coefficient,  it 
has  been  agreed  to  enclose  the  former  in  a  parenthesis ;  thus  we 
write 

VdtrK  _  du    dp      dti  ^  Fdn  "1  du   dp  du 

Lc/iJ       dp    dx      dx  '    '        ""  LdxJ       ~  dp   dx  dx 

Here  again  there  is  a  necessity  for  caution,  so  as  not  to  confound 

du 

—  •  dx  with  du ;  the  former  being  only  a  part  of  the  change  im- 

O-X 

parted  to  u  by  a  change  in  x,  while  the  latter  is  the  symbol  of  the 
entire  change. 

Cor.  if  there  were  given      u  =1  ^{Pt  ^t  ^0 
where  p  and  q  are  functions  of  a;,  then 

\d^r\       du    dp       du    dq       du 
\jix\       op    dx        dq    dx       dx 

and  similar  expressions  would  apply  if  there  were  a  greater  numbei 
of  functions. 

EXAMPLES. 

56.  1.     u  =  sin—^  (p  —  q),     where    p  z=  Sx    and     q  =  4a^. 

^P       ^1  -  (jo  -  qf    ^2       yi  -  (p-qf  dx         '    dx 
dn  __du    dp       du    dq  3  -  V2x'^ 


dx       dp    dx        dq    dx       y'i_(^)_o)2 
3  -  VZx'  3 


2.  u  =:pq^  where  p  =  e*,  and  q  =  x^  —  'ix^  -h  ISx^  —  24a:  +  24 
du  du  dp  dq        .   o        ,  ^  „       ^ . 


IMPLICIT  FUNCTIONS.  73 

du       du    dp       du    do  ^  • 

.  • .  —  =  —  '  -^  -\ -i  =  e*  .  ar*. 

dx       dp    dx        dq    dx 

x^p^        x^p       X* 
t.  *''  =  ~4 8  ""^32'      ^  p  =  \ogx. 

du  _  x*p       X*      dp  _\      du  _    3  2       x^p    I    ^^ 

=r  a;^  (log  a;)  2. 

4.      «  =  —  ^         where  •  p  =:  a  sin  a:,      and      q  =  cos  a;. 

c?w  f**^         .c?'<  c°*         dp  dq 

-—  =  — -,     -^  = T— — tj     -f-  =  a  cos  a;,     -—  =  —  em  ar, 

rf/t>       a^  +  1      (/g  a-^  -\-  V     dx  '      dx  * 

dx  ~       a2  4-  1 

\.dxA  ~  dp    dx        dq    dx       dx 

gax 

=  — — - — -  (a  cos  a;  +  sin  a;  +  a^  sin  x  —  a  cos  a;)  =  c°*  sin  x. 
a^  +  1  ^  ' 

Differentiation  of  Im/plicit  .Functions, 

57.  In  the  various  cases  hitherto  considered,  we  have  supposed 
.he  function  to  be  given  explicitly  in  terms  of  the  variable.  It  is 
now  proposed  to  establish  rules  for  differentiating  implicit,  functions. 

Prop.  Having  given  F(x,  y)  =  0,  to  form  the  differential  coeffi- 
cient —-  without  solving  the  equation. 

Put  u  =  F (x^y):  then  u  will  be  a  function  of  x  directly,  and 
also  indirectly  through  y. 

[dn~\       du    dy       du 
dxj~dy    dx       dx 


du 

dy 

dx 

dx  ~ 

du 

dy 

,74  DIFFERENTIAL    CALCULUS. 

But  since  u  remains  constantly  equal  to  zero,  the  total  differen- 
tial coefficient  oi  u  with  respect  to  x  must  be  equal  to  zero  also. 


du    dy       da       ^  , 

-; r-  +  -7-  =  0,         whence 

dy    dx       dx 


Thus   it   appears  that  we  must  form  the  partial  differential  co- 
efficients   -7-   and  T",  then   divide  the  former   by  the  latter,  and 
dx  dy  J  1 

prefix  the  negative  sign  to  the  quotient. 

Ex.  1.  y2  _  2axy  -f  a;^  —  6^  ==  0,  to  form  the  differential  coeffi- 
cient of  y  with  respect  to  x. 

u  —  y"^  —  2axy  +  a;^  _  ^2^    _^_  _  ^ay  +  2a:,     — =  2y  —  2aa?. 

dy  —  2ay  -\- 2x  _  ay  —  x 

'    '   dx  2y  —  2ax         y  —  ax 

2.  Given  x"^  -f  Zaxy  -\-  y^  =  0^  to  form  the  1st  and  2d  differen- 
tial coefficients  of  y  with  respect  to  x. 

.  • .    M  =  a;3  +  Zaxy  +  y^     -^  -  3x^  +  3ay,     -^  =  Sax-\-  Sy», 

dy  _       a;2  -(-  ay 
dx  ax -{- y"^ 

Put  J-=  Pi ',   then  pi  will  be  a  function  of  x  directly,  and  also 
indirectly  through  y. 

d 
'  ix^  ^  Ldx  J~  dy 


d^y  _  Vdp^  "I  _  dp^    dy      dp■^ 
'~LdxJ~  dy     dx       dx 


But 

^  _  -«(«^+2/')  +  2y(a:^4-ay)   dp^  _  -2x{ax-hy^)-^a(x^ -^  ay)^ 
dy  "~  {ax  +  2/2)2  '  ^^^  —  (^^j  ^  ^2)2 


IMPLICIT  FUNCTIONS.  75 

Hence  by  substitution  and  reduction 


\      ax  -\-  //7 


dx^  [ax  +  2/^)^  \      ax  -{-  y^f  {ax  -f-  y'^Y 

2a^xy  —  2xy[x^  -\-  Saxy  -f-  y^)  2a^xy 

"  {ax  +  y^)^  ~  {ax  +  y'^Y 

58.  Since  it  is  possible  to  form  the  successive  differential  coeffi- 
cients of  y  with  respect  to  x^  without  solving  the  given  equation,  it' 
will  be  possible  to  expand  y  in  terms  of  x  by  Maclaurin's  Theorem. 

1.  Given  y^  _  3^  _|_  ^^  —  0, 

to  expand  y  in  terms  of  the  ascending  powers  of  x. 

^         ^  '     dx         '    dy         ^^         '  dx      3(1— y^) 

Expanding  the  last  expression  by  actual  division,  we  have 

|=J(l+,2  +  ,.  +  ^,) 

.  • .  §  =  ^  (2y4-  42/^ -f  62/5-f  &c.)  ^  =  1  (2y  +  63/3+  l2y^+  &c.) 

^=.1-  (40  +  1080^/2  4-  &c.)~  =  ^(40  4-  1120/^4-  &c.)&c 
But  when  a;  =  0,  [y]  =  0, 

•  L^J^i'  Lrf^J=^'  L;z^J=3^'  L^J=^'  Ud^y^'*"* 

.'.By  substitution  in  Maclaurin's  formula, 


I 
tQ  t)IFFERENTIAL    CALCULUS. 

2.  To  expand   y  in   terms  of  the  descending  powers  of  ar,  from 

the  relation 

ay^  —  x^y  —  ax^  =  0. 

Put  x^  =  -:         then         ayH  —  y  —  a  =  0. 

V 
o  ^'*  3         (^^'  o       2  1  ^.y  ^^^ 

,*.  u—ay^v—y—a,   -r-  —  ay^.     ~r-  =  oay^v  —  1,     -j-  =  i — x— 

-^         ^        ^    dv  dy  *  '     c/y        1  —  I^t/</2« 

^  = (1  -  ^..y^,^ '  ^^•'  ^«- 

But  when     t;  =  0,  [y]--a,      [J]=-<      [^]  =  -  Ca^  &a 

,'.  y  =  -a  —  - j— -  &c. 


or  by  rephicing  v  by  — , 


1  1.2 


The  use  of  this  method   is  much   restricted   by  the  great  labor 
usually  required  in  forming  the  successive  differential  coefficients. 


CHAPTER    VII. 

ESTIMATION    OF    THE    VALUES    OF    FUNCTIONS    HAYINO    THE 
INDETERMINATE    FORM. 

59.  It  frequently  occurs  that  the  substitution  of  a  particular  value 
for  a  variable  a;  in  a  fractional  expression  will  cause  that  expression 

to  assume  the  indeterminate  form  -•      Such  expressions  are  often 

called  Vanishing  Fractions,  and  they  may  be  regarded  as  limits  to 
the  values  of  the  ratios  expressed  by  these  fractions,  when  the 
variable  value  of  x  is  caused  to  approach  indefinitely  near  to  some 

particular  value. 

X*  —  I 

Thus  in  the  example  u  =  — — ,  the  value  of  which  can  usually 

be  determined  when  that  of  x  is  given,  by  a  simple  substitution,  we 

find  that  it  assumes  the  form  -  when  x  =  1.      But  the  value  of  u 

is  even  then  determinate ;  for  if  we  divide  the  numerator  and  de- 
nominator of  the  fraction  by  a:  —  1,  before  making  a?  =  1,  we  get 

x^  -\-  x^  -{-  X  -\-  1 
"~       x^-i-  x+1 

as  a  general  value  of  w,  and  this  becomes 

1  4-  1  4-  1  -f  1       4 


-       w 


hen       X  =  I. 


1  +  1  +  1  3 

Here  we  see  plainly  that  it  is  the  presence  of  the  common  factor 
a?  —  1  in  the  numerator  and  denominator  which  causes  the  fraction 
to  assume  the  indeterminate  form.     In  this,  and  in  all  similar  cases, 


7.^  DIFFERENTIAL   CALCULUS. 

the  removal  of  the  common  factor  serves  to  determine  tlie  value 
of  u.  But  it  usually  occurs  that  the  discovery  of  this  factor  is 
attended  with  considerable  difficulty,  and  hence  the  necessity  of 
some  more  general  method  by  which  to  estimate  the  values  of  frac- 
tions which  assume  the  indeterminate  form  -,   when  the  variable  x 

tal\es  a  particulai  value.  Such  a  method  is  readily  supplied  by  the 
Differential  Calculus. 

It  should  be  observed,  however,  that  there  are  other  indeterminate 

forms  besides  -,  such  as  the  following : 


0' 

00  X  0,     00  —  00  ,     00,     00  0.     1- 


00  .--_..  .         -  + 


00 

each  of  which  will  be  considered  in  succession. 

60.  Pro}^  To  determine  the  value  of  a  function  which  takes  the 

form  -  for  a  particular  value  of  the  variable. 

Let  u  =  —  =  be  a  function  which  takes  the  form   -   when 

Q         (px  0 

X  =  a;  that  is,  let  Fa  =  0,  and  cpa  =  0 :  let  it  be  proposed  to  find 
the  particular  value  [w]  assumed  by  u  when  x  =  a. 

Suppose  X  to  take  an  increment  h,  converting  w,  P,  and  Q 
into  t«i,  Pj,  and  Q-^,  respectively,  and  let  Pj  =  F  (x  -{-  h)  and 
Q^  =.  (p  [x  -\-  h)  be  expanded  by  Taylor's  Theorem  :  then  denoting 
the  successive  differential  coefficients  Fx  by  F^x,  FnX,  &;c.,  and  those 
of  (px  by  cpiX,  (p^,  &c.,  we  have 

P.      Fjx^k)      Fx  +  F^l^F^x^^^F^x^^^^. 
01*  when  x  =  a 

(pa  +  <Pi«  Y  +  92«  ]— 2  "'"  ^^^  1    2   3  "*"  ^^' 


FUNCTIONS  HAVING    THE   INDETERMINATE  FORM.  79 

But  by  hypothesis,  Fa  —  0,  and  9a  =  0.  .  * .  Omitting  the  first 
term  in  the  numerator  and  denominator,  and  then  dividing  each  by 
h.  we  get 

•"  =  — T- ¥—~  •■•(') 

Now  making  A  =  0,  we  convert  w^  into  [w],  and  thus  obtain 
[u]  =  — L. 

Hence   it   appears   that,  in  order   to  determine  the  value  of  a 

function  —  which  takes  the  form  -  when  x  =  a,  we  must  replace 

(px  0 

Fx  and  (px  by  the  values  of  their  first  differential  coefficients,  and 
then  make  x  =  a  in  each. 

It  will  sometimes  occur  that  this  substitution  will  reduce  to  zero 
both  F^a  and  cp^a^  in  case 

r  -,        F.a     0  .         .11       -.  .     , 

If    = =  -  remams  still  undetermmed. 

•-  -^        (p-^a      0 

we  then  omit  F^a  and  cp^a  in  equation,  (I)  and  divide  the  numerator 
and  denominator  by  --^,  thus  obtaining 

F^a  +  F,a^-h&;c, 
u,  = 1 ...(2) 

<p^a  4-  93«  3  +  &c. 

which  becomes  \u]  =  — ^ 

when  ^  =  0.  ' 

.  * .  when  the  first  diff*erential  coeflScients  both  reduce  to  zero,  they 
must  be  replaced  by  the  second  difl'erential  coefficients.     If  Fza  and 


/ 


80  DIFFERENTIAL   CALCULUS. 

<p2«  both  become  zero  also,  we  omit  them  in  (2),  then  divide  by 

h 

-,  and  finally  make  h  =  0,  obtaining 

u 

[w]  =  -2-. 

And  since  the  same  reasoning  may  be  extended,  we  have  the  fol- 

lowinf?  rule  for  findino;  the  value  of  \u\  =z  —  =  -,  viz. : 

^  ^        (pa       0 

Si(hstitute  for  Fx  a)id  (px  their  Jirst,  second^  third^  cC'c,  differential 

coefficients^  and  make  x  =:z  a  in  each  result^  until  a  pair  of  coefficients 

is  obtained,  both  of  which  do   not  reduce  to  zero ;  the  fraction  thus 

found  will  be  the  true  valUe  of  [u]. 


EXAMPLES. 

61.  1.  u  =2 —  =  -       when       ic  =  1. 

X  —  i        0 

Fx  =  x^  —•  1,     and    (px  =  x  —  I.     .  • .  F^x  =  5x\     and     (p^x  ■=  1. 

.  • .  F-,a  =  5,     (p.a  =  1,     and     [wl  =:  — i-  =  -  =  5. 

*-  -*         (p^a        1 

This  result  is  easily  verified  by  division,  before  making  :r  =  1 ;  thus 
by  actual  division 
x^  -  1 


X  —  1 


=z  x^  -[-  x'-^  -\-  x^  -\-  X  -^  I  =  5     when     x  =  I, 

«*-6*       0        .  . 

u  = =  -     when     x  z=  0. 

X  0 


F.x        ]o<f  a.a*  —  loff  b .  b'  F.a       ,  ,       ,        r  -i 

^  =  -= r-^ .•.^  =  loga-Iog6  =  [«]. 

This  result  is  easily  verified  by  expanding  a*  and  b*. 

a*  —  b" 


Thus 


J  Or  X  X 

1  +  log  a  .  J  +  log^a  _  -j-  &c.  -  1  -  log  6.  J  —  log^A  •  -v^  -f  &c. 


FUNCTIONS  HAVING  THE  INDETERMINATE  FORM.  81 


X 

.  • .  tt  =  log  a  —  log  6  '{-  - — -  (log^a  —  log26)  +  <S20. 

.  • .  [w]  =:  log  a  —  log  6       by  making       a;  =  0. 

a  —  Va2~—  x^      0 

3,  «  = 5—- =  -       when        x  =  0, 

x^  0 

^  _  ar(a2  -  a:^)"*  ;        /   F,a  _0 
(p-^x  ~  2x.  '       '    *  9ja        0 

Here  the  first  differential  coefficients  prove  equal  to  zero,  and 
therefore  they  must  be  replaced  by  the  second  differential  coeffi- 
cients.    But 

F^x        (a^  -  0:2)"*  +  a:2(r/2  _  a:2)"* 


92^                              2 

92^            2           2a       ^   ' 

4L 

aa;2  +  ac2  —  2af^       0 

6.c2  —  •^6c^  -f  6c2       0 

FyX        2ax  —  2«c         ^    i^ja       2ac  —  2ac       0 
9ia:  ~^  —  2bc       '    '  9^0  ~  26c  —  26c  ~  0 

Then 

""^      2a             ii^2«       2a       a 

92^;  ~  2<i        •    '  92^  ~2b~  b~  ^^^' 

5. 

x^  —  ax^  —  a'^x  +  a^       0 

x^  —  a^                 0 

i^'io;  _  3ar2  _  2^^y,  _fj2             F^a       0 
(PiX                  2x                  *    '  9ia  ~  2a           —  L  J* 

6. 

ax  —  n2                    0        , 

''~a^-2a^x  +  2ax^-x*-0     ''^''^     "^  ^ ''• 

F,x 

_             a  ~  2a;                      ^    i^^a                   a  —  2a 

iPlX 

—  2a3  4-  Gax^  —  4x^       '    '  cp^a  ~  —  2a3  +  Qa^  —  4a^ 

or 

F,a            a                      r  T 
9ia            0                      '-  -^ 

82 
7. 


DIFFERENTIAL   CALCULUS. 

] 
u  = 

(1 

log  a;          0        , 

— i  ~  n    ^^^"    X  =  1, 

F,x 

1                  i 

a;                      2(1  -  xf 

9ia? 

in    .r*~ 

(p-a  1 

8.      w  =  -, :—  =  ^      when     x  =z  a,     {s  being  an  integer.) 

Diiferentiating  s  times,  we  get 


9^        s{s-l){s-2) 3.2.1 

•  *•  ^  ~  s{s-  l){s-2) 3.2.1  ""  ^'^^' 

^  tan  a;  —  sin  a;       0        ,  ^ 

9.  t^  = r-o =  t:     "when     x  =  0. 

sm-^x  ■   0 

J^ja;      sec^a;  —  cos  x        ;     F^a  _  sec^O  —  cos  0  _  0 
<p,a;  ~  3  sin^a; .  cos  2;'      '    '  cp-^a  ~  S  sin^O  .  cos  0  "~  0 

i^2^       2  sec%  .  tan  a;  4-  sin  x 
(P2X  ~~  6  sin  X .  cos^a:  —  3  sin^a; 

F^a  _  2  sec^-0  .  tan  0  +  sin  0  _  0 
'  * '    9^  ~  6  sin  0. cos^U  —  3  sin^O  ""  0  ' 

FnX  4  sec^a: .  tan^ar  +  2  sec*a;  +  cos  x 


(p.^       6  cos^a:  —  12  sin^a: .  cos  a;  —  9  sin^a; .  cos  x 

F^a  __        4  sec'^O  .  tan^O  +  2  sec^O  -f-  cos  0 3  _  1       .-  - 

'"    (p'^a  "  6  cos^O  —  12  sin^O .  cos  0  —  9  sin20 .  cos  0  ""  6  ~  2  "  '  "-^ 

62.  The  method  just  explained  and  illustrated,  ceases  to  be 
applicable  when  we  obtain  a  differential  coefficient  whose  value 
Dccomes  infinite  by  making  x  ^=^  a ;   tor  such  a  result   shows  the 


FUNCTIONS   HAVING  THE   INDETERMINATE   FORM.         83 

impossibility  of  developing  the  corresponding  function  F  {x  -\-  h)  by 
Taylor's  Theorem,  for  that  particular  value  of  ar,  and  therefore  the 
process  founded  on  such  development  fails. 

The  expedient  adopted  in  such  cases,  is  that  of  substituting 
a  -{•  h  for  X,  then  expanding  numerator  and  denominator  by  the 
common  algebraic  methods,  then  dividing  numerator  and  denomi- 
nator by  the  lowest  power  of  h  found  in  either,  and  finally  making 
A  =  0.     A  few  examples  will  illustrate  this  method. 

(^2  _  ^2\^  Q 

63.   1.  u  = o  =  t:  when  x  =a. 

(a  —  X) 

Here  the  first  differential  coefficients  reduce  to  zero,  and  all  suo 
ceeding  coefficients  become  infinite  when  x  =  a.  We  therefore  put 
a  -{-  h  for  X  and  expand. 


•• 

(a2  _a2-2ah-  A^)*        (2a  -f  h)^,(-  A)* 

=  (2a  +  A)*=  (2a)^+  |  (2a)*  A  +  &c. 

.-.    M=(2«)f 

2 

^/x  —  -J a  +  -v/^  —  a      0     , 

^x'^  —  a?                0 

Put 

♦                          a  -{■  h     for     X 

• 

(a  +  A)*  _  a*+  (a  +  A  -  a)*      h^+la'^h  -  &a 

(a2  +  2  aA  4-  A2  -  a^)^                ^^(2a  -f  ^r 

1 -|-ia"*A*&c. 

(2a)*-f  i(2a)~*A&c. 

(2a)* 

84  DIFFJaiENTIAL   CALCULUS. 

Remarl\  This  method  may  be  used  even  in  those  cases  to  which 
(he  method  of  differentiation  is  applicable.     We  will  now  consider 

the  other  indeterminate  forms. 

P     Fx 

64.  Prop.  To  find  the  value  of  the  function  w  =  -— •  =  — .  which 

Q       (px 

assumes  the  form  ^  when  ar  =  a. 

GO     _^__ 

Put  P  =  —  and   Q  —  —     Then  we  have 

p  q 

1 

P        5'       ^     u 
u  z=  ^  =z  -=:  -  when  a?  =  a. 

1       />      0 

Thus  the  function  being  reduced  to  the  ordinary  form  -,  its  value 
may  be  found  by  the  methods  already  explained. 


Now 

since     p  - 

1 

dp 
dx~ 

1      dP 

P2       dx     ' 

=  — • 

F,x 

(Fxf 

And 

similarly 

dq 

cp,x 

' 

dx 

(^xf 

9i« 

.-.  [u\. 

Fa 

~"  (oa 

-    F,a 
{Faf 

9i« 
F,a 

.-.  1  = 

Fa 
X 

(pa 

;  47-    whence = 

Jl^a                     cpa 

F,a 
'  (p^a 

=  [4 

Hence  it  appears  that  the  ordinary  direct  process  of  substituting 
for  numerator  and  denominator,  their  first  differential  coefficients 
will  apply  when  the  function  takes  the  form  ^.  But  since  when 
P  =  00  and  ^  =  ao ,  their  differential  coefficients  will  also  be  in- 
finite, the  reduced  fraction  will  still  take  the  form  ^.  and  therefore 

'  CO  ' 

will  not  serve  to  determine  the  true  value  of  u^  unless  we  can  dis- 
cover a  factor  common  to  the  numerator  and  denominator,  or  can 
trace  some  relation  between  the  numerator  and  denominator  of  the 
•^'^w  fraction,  which  will  facilitate  the  determination  of  its  value. 


FUNCTIONS  "HAVING  THE   INDETERMINATE   FORM.  85 

65.  Prop.  To  find  the  value  of  the  function  u  =z  Px  Q  =^Fx  X  <i;>ii 
which  takes  the  form  oo  X  0  when  x  =  a. 

Put  P  =  —     Then  u  =  —  =  ^    when    x  =  a,  the  common  form. 
p  p       0 

.,        .  1  ,  dp  I     dP  F.x 

Nowsmce^  =  -,     we  have     -^-__=-^^^^ 

But  since  when  P  =  i^a:  =:=  oo ,  its  differential  coefficients  will  aisc 
be  infinite,  the  value  of  u  will  take  the  form  ^,  unless  the  infinite 
factor  should  disappear  by  division. 

66.  Prop.  To  find  the  value  of  the  function  u  z=P  —  Q  =  Fx  —  cpx^ 
which  takes  the  form  qo-qo  when  x  =:  a. 

Put  P  =  -  and   Q  =  -'     Then 

P  9 

1        1       9—P       0         , 

u  = = =  -      when^    x  z=  a^ 

p       q  pq  0  ' 

and  the  value  is  to  be  found  by  the  ordinary  method. 

67.  Prop.  To  find  the  value  of  the  function  u  z=  fI=  (Fxf  which 

+  00 

takes  either  of  the  forms  0°,  oo  o,  or  1        ,     when      x  =  a. 

1st.  Let  the  form  he  u  =  0°.     Passing  to  logarithms  we  have 
log  u  =  Q.  \ogP  =  (px.  log  {Fx) 
'  •   Uog  u]  =  (pa.  log  {Fa)  =  —  0  x  <x> , 
which  is  one  of  the  forms  already  provided  for. 
Thus,  having  found  log  u,  we  have  u  —  €^°s  «. 
2d.  Let  the  form  be  i^=oo  «.  Then  log  u=:Q.  log  P-z(px.  log  (Fx), 
•  ■  •   ['og  u]  =  (pa.  log  {Fa)  =  0  X  oo . 
a  form  already  considered. 


86  DIFFERENTIAL    CALCULUS. 

3d.  Let  the  form  be   1*"  . 

Then  log  u  =  Q  log  P  =  (px .  log(^a:). 

.  • .   [log  u]  =  (pa.  log  (Fa)  =  db  oo  X  0, 
and  the  form  is  still  the  same. 


EXAMPLES. 


68 

Here  Fx 


.  1.     u  =  {1  —  x).teLnlx'-\=  0  X  (X)      when     a;  =  1. 
=  tan|a;'-j         and         (px  =  l^x, 
• .    F^x  =  '^  sec^^a;  •  0,     (p,x  =  -  1, 

2-sec^(lX2) 
(l-cos=l*)=H. 


ir         1  'n' 


1 

2.     w  =  e''".  sin  ar  ==  00  X  0     when  a:  =  0. 

L  1     L 

Fx  =  e'  y     (px  =  sin  a;,     .  * .  F^x  = -e' ,     cp^x  =  cos  x. 

Here  the  function  still  takes  the  form  oo  X  0 ;  but  the  true  value 

\_ 

is  easily  found  by  expanding  e". 

For        >..-(! +l+^_  +  .^-^3  +  &o.)«> 
=  ^.  +  ,  +  _i_  +  _L_  +  &c 
.-.       e*  X  02  =  «  =  [«]. 


FUNCTIONS   HAVING  THE  INDETERMINATE  FORM.  8f 


a:"       00 

3.  M  =  —  =  —     when     a:  =  GO  . 

e*  GO       , 

Differentiating  n  times,  we  get 

^  n(»-l)(H-2)....3.2.1  ^ 

L    J  goo 

loi^a:       GO         , 

4.  «  =  =  —     when     a;  =  oo . 

a;"         QO 

...    M  =  :?>  =  _i_=o. 

6.     u  = :; =  c»  —GO      when    ar  =  1. 

I  —  X       I  —  x^ 

I       \  1        l-a;2 

1(1    _a;2)   _(1    -x) 

= =  -    when  X  z=l, 

1 

r  T         2  1. 

J.     u  = 7  —  :; =  00  —  (©     when    a;  =  !• 

a;  —  1       log  a: 

I        x  —  \  , 

M  (^  ■"  V  ^^*g^        ^ 


SS  DIFFERENTIAL   CALCULUS. 

_  log  1  4-  1  -  3  _  0 

•  *  •  L''J  ~  i7jg  1  -f  nil  ~  0  * 

Differentiating  numerator  and  denominator  of  the  value    of  m  l 
second  time,  and  making  x  =  1,  we  get 

r  1       ^  ^ 

7.  u  :=  X'  =  0°     when     x  =  0. 

log u  =  X.  log  a;  =  —Ox  <» ,     when     ar  =  0. 

,              log  a?            00       ,  ^ 

or,  log  u  =     7     = when  ar  =  0. 

°  1  00 

ar 
Then  Fx  =  log  a:,        and        (pa;  =  -• 

1 
.  • .   — ^  = =  -—  ar.       .  • .    flog  u]  =  — ^  =  0,   and    [u]  =  1 

8.  i*  =  ar"»*  =  0°,     when     x  =  0. 

Since  =  I  when  x  =  0,    .  • .  a-^'^  '  =  x'  =  1  when  a;  =  0. 

And  similarly     sin  a:»'°  *  =  a:*  =  1   when  x  =  0. 
Again,  since        sin  x .  log  a:  =  sin  ar .  log  x  .  log  e. 

.-.   a;8iii*  _  gsinx.log*  _  1    ^h^jn    a;  =  0. 
.  • .    sin  X  .  log  X  =  0,  when  a:  =  0. 
And  similarly     siji  x .  log  sin  a;  =  0  when  a:  =  0. 

9.  u  =  cot  a;"'"*  *  =  oo  <>  when  x  =  0. 

cos  a; 
log  w  =  sin  ar .  log  -r —  =  sin  x  (log  cos  x  —  log  sin  x). 

=  0  —  0  =  0     when     a:  =  0,         .  • .  [w]  =  1. 


FUNCTIONS  HAVING  THE  INDETERMINATE  FORM.  89 

1 
10.     w  =  (1  -f  nx)'  =  1"     when     x  =  0. 

loff  (1  +  nx)       0        , 

loff  u  =  — ^-^^ =  -     when    x  =  0. 

°  X  0 

.  •.  By  differentiation,    [1  >g  w]  =  -  =  n,     and     [«]  -.=  e*. 

This  result  is  easily  verified ;    for  by  expanding  (1  +  nx)'     by 
the  binomial  theorem,  we  obtain 


which  series  is  the  expansion  of  e". 

11.  w  =  (cos  aar)  =  l°°j     when     x  =  0. 

loff  cos  ax      0       . 
log  u  =  cosec^ca; .  log  cos  ax  —  —^- =  -     when     x  =  0, 

Put  log  cos  ax  =  Fx       and       sin^ca?  =  (par. 

,  • .  F-^  =  —  a.  tan  ax,     (p^x  =  2c .  sin  ex.  cos  ex  =  c.  sin  2  ex, 

/>_  0 

9ia  ~  0 
Diffeientiating  again  we  get 

F^  =  —  a^ .  sec^aar,     <p2^  =  ^^^  •  ^^^  ^^^' 
.  • .  — ^  = •     .  • .  ImI  =  e  ^ 


CHAPTER  VIIT. 

MAXIMA    AND    MINIMA    FUNCTIONS    OF    A    SINGLE    VARIABLE. 

69.  If  u  be  a  function  whose  value  depends  on  that  of  a  variable 
a*,  so  that  u  =  Fx,  and  if,  when  x  takes  a  certain  value  a,  the  cor- 
responding value  W|  of  u  be  greater  than  the  values  which  immedi- 
ately precede  and  follow  it,  then  the  value  Wj  is  called  a  maximum  ; 
but  if  the  intermediate  value  be  less  than  those  which  precede  and 
follow  it  immediately,  the  value  w^  is  said  to  be  a  minimum. 

Suppose  for  example  that  when  x  =  a,  the  general  value  w  =  Fx 
becomes  Wj  =  Fa,  that  when  x  =  a  ±i  h  u  becomes  Wg  =  F(a  -\-  h\ 
or  y/g  =  F((i  —  A),  and  suppose  that  for  some  small  but  finite  value 
of  h,  and  for  all  values  between  that  and  zero,  the  corresponding 
values  of  both  V2  and  Wg  shall  be  less  than  Wj,  then  will  u^  be  a 
maximum;  but  if  U2  and  u^  be  both  greater  than  u^^  then  the  latter 
will  be  a  minimum. 

70.  In  order  to  discover  the  conditions  necessary  to  render  a 
function  {u  =  Fx)  either  a  maximum  or  minimum,  the  following 
principle  will  be  established. 

Prop.  In  any  series  Ah'^  -f-  -Bh^  +  Oh*^  +  &c.,  arranged  accordmg 
to  the  positive  ascending  powers  of  h,  a  value  may  be  assigned  to  h 
so  small  as  to  render  the  first  term  ^A*,  (which  contains  the  lowest 
power  of  A),  greater  than  the  sum  of  all  the  succeeding  terms. 

Proof.  Assume  A  >  Bh^-'^  +  Ch'^"'^  -+■  <S2C.,  a  condition  always 
possible,  since  by  diminishing  h  the  second  member  may  be  ren- 


MAXIMA  AND  MINIMA.  91 

dered  less  than  any  assignable  quantity.  Multiply  each  member  by 
A« ;  and  there  will  result  Ah*^  >  Bh^  +  Ch'^  -}-  &;c.,  as  stated  in  the 
enunciation  of  the  proposition. 

Cor,  The  value  of  h  may  be  taken  so  small  that  the  sign  of  the 
first  term  shall  control  that  of  the  entire  series. 

71.  Prop.  To  determine  the  conditions  necessary  to  render  a 
function  w  of  a  single  variable  x^  either  a  maximum  or  minimum. 

Let  u  =  Fx,  and  suppose  x  to  receive  successively  an  increment 
and  a  decrement  h.     Then  developing  by  Taylor's  Theorem,  we  get 

rr/      ,    X.X         ^     .    dFx    h     ^     d-'Fx      h?      ,    d?Fx         h?        ,     , 

«,=^(.+A)=^.-f^.;^  +  ^.^  +  -^.j-^+&c.     (1) 

^,       ^,      ^       dFx  h      dP-Fx    h?        d?Fx      A3 
«3=^(.-A)=^.-  — .-  +  _._.-— .^_  +  &c.    (2). 

Now  in  order  that  Fx  may  exceed  both  F(x  4-  K)  and  F{x  —  h)^ 
it  is  obviously  necessary  that  the  algebraic  sum  of  the  terms  suc- 
ceeding the  first  term  in  each  of  the  series  (I)  and  (2)  shall  be 
negative ;  that  is,  we  must  have  by  employing  the  usual  notation, 

•^1^- Y  +  ^2^172  +  ^^"^TTsTs"  +  &<=•<<>••••  (3), 

/?  7/2  7;  3 

and    -F,x-j  +  F^-'YTi-  ^^^rTaTs  +&<=.<  0  ...  .  (4). 

Now  the  sign  of  the  first  term  in  each  of  the  series  (3)  and  (4) 
will  control  that  of  the  entire  series  when  h  is  taken  sufficiently 
small,  and  since  the  first  terms  of  (3)  and  (4)  have  contrary  signs, 
it  is  impossible  that  both  of  these  series  shall  be  negative,  so  long. as 

the  term  F^x .  -  has  a  finite  value.  Hence  the  first  condition  neces- 
sary to  render  Fx  a  maximum  is  that  F^x .  --  =  0,  or  since  h  is  finite 

^...^^  =  0....(5). 


92  DIFFERENTIAL   CALCULUS. 

Now  oniitting  the  first  terms  of  (o)  and  (4),  we  have 

and  Fr,x  _  -7^3a;.-j-^-^  +  &c.  <  0  .  .  .  .  (7). 

The  signs  of  the  series  ((i)  and  (7)  will  be  controlled  by  those  of 
their  first  terms,  which  Urnis  have  the  positive  sign  in  both  series; 

and  therefore  each  series  will  be  negative  when  F2X  - — -  is  an  essen- 
tially negative  quantity,  or  when  i^g-r  ^^  essentially  negative,  (since 
- — -  IS  always  positive). 

Thus  the  two  conditions  which  usually  characterize  a  maximum 
value  of  Fjc  are 

dFx       ^  ,       cf^Fx 

^  =  0,       and       —  <0. 

On  the  contrary,  when  Fx  is  a  minimum,  we  must  have  Fx  less 
than  F{x -{- h)  and  F{x  —  h),  and  therefore  by  a 'similar  counse 
of  reasoning,  the  necessary  conditions  are 

dFx       ^  ^       d^Fx       ^ 

__.^0,       and       —  >0. 

The  conditions  here  obtained  are  those  usually  applicable  :  the 
exceptions  will  now  be  considered. 

72.  The  results  obtained  in  the  last  proposition  indicate  the 
following  as  the  ordinary  rule  by  which  to  discover  those  values  of 
the  independent  variable  x,  which  will  render  any  proposed  function  u 

a  maximum  or  minimum. 

du 
1st.  Form  the  first  differential  coefficient -7—,  place  its  value  equal 

to  zero,  and  then  solve  the  equation  thus  formed,  obtaining  the 
several  values  of  x. 


MAXIMA  AND   MINIMA  93 

2d.  Form  the  second  differential  coefficient  -— -,  and  substitute  for 

J?,  in  the  value  of  that  coefficient,  each  of  the  values  found  above. 

d/'-u 
Then  all  those  values  of  x  which  render  -—  negative,  will  corres- 

CLX 

pond  to  maximum  values  of  u  ;  but  those  values  of  x  which  render 

dhi 

-—  positive,  will  correspond  to  minimum  values  of  u.     And  when 

ax 

the   proper   values  of  x  have  been   ascertained,  the  maximum  or 

minimum    values  of  u   are   found   by  simple   substitution   in   the 

•equation  u  z=.  Fx. 

73.  1.  In  the  application  of  the  preceding  method,  it  may  occur  that  a 

value  of  ar,  obtained  by  making  — -  =  0,  will,  when  substitued  in  -r-— , 

dx  dx^ 

cause  that  coefficient  to  reduce  to  zero  also.     In  that  case,  the  signg 

of  the  series,  (6)  and  (7),  in  the  last  proposition,  will  depend  on  the 

terms  which   contain   the  third   differential   coefficients ;    and   since 

these  terms  have  contrary  signs  in  the  two  series,  the  value  of  x 

which  renders  -—  =  0,  and  -—-  =  0,  cannot  render  it  either  a  maxi- 
dx  dx^ 

mum  or  minimum,  unless  it  should  happen  to  render  -—  =  0  also. 

When  this  occurs,  we  must  examine  the  sign  of  the  fourth  differential 
coefficient,  which  now  controls  the  sign  of  each  series,  and  if  this  be 
negative,  the  value  of  u  will  be  a  maximum ;  but  if  positive,  a 
minimum. 

And  since  the  same  reasoning  could  be  extended  when  other  differ 
ential  coefficients  reduce  to  zero,  we  have  the  following  more  general 
rule  for  the  discovery  of  maximum  and  minimum  values  of  a  func- 
tion of  a  single  variable. 

1st.  Form  the  first  differential  coefficient,  place  its  value  equal  to 
zero,  and  deduce  the  corresponding  values  of  x. 

2d.  Substitute  each  of  these  values  in  the  succeeding  differential 
coefficients,  stopping  at  the  first  coefficient  which  does  not  reduce  to 


^ 


94 


DIFFERENTIAL  CALCULUS. 


zero.  If  this  coefficient  be  of  an  odd  degree,  the  corresponding  value 
of  u  will  be  neitner  a  maximum  nor  a  minimum ;  but  if  it  be  of  an 
even  degree,  the  value  of  u  will  be  a  maximum  or  mininum,  accord- 
ing as  the  sign  of  that  coefficient  is  negative  or  positive. 

The  annexed  diagram  will 
illustrate  the  fiict  that  the 
same  function  may  have  sev- 
eral maximum  and  several 
minimum  values ;   and  that       0    P      Q        R  S  X 

one  minimum  may  exceed  another  maximum.  Thus,  if  the  curve 
CDEFGH  be  the  locus  of  the  equation  y  —  Fx,  then  will  DQ  and 
FS  represent  maximum  values  of  the  ordinates  y,  while  (7P,  ER, 
and  GX  \n\\\  be  minimum  values  of  the  same.  Also  the  minimum 
6^ A" exceeds  the  maximum  DQ. 

74.  The  substitution  of  a  value  x  ■=  a.  derived  from  the  equation 

^  =:  0,  in   the  succeeding  differential   coefficients,  will  sometimes 

cause  the  first  of  these  coefficients  w^hich  does  not  reduce  to  zero,  to 
become  infinite. 

This  happens  only  when  the  development  o^  F  (x  -{•  h)  in  the 
ordinary  form  (by  Taylor's  Theorem)  is  not  possible  for  that  parti- 
cular value  of  X.  We  must  then  find  by  other  methods  (such  as 
algebraic  development)  the  true  value  of  the  term  which  cannot  be 
obtained  by  Taylor's  Theorem.     If  it  be  found  to  contain  a  power 

JJL  1 

of  A,  which  will  change  sign  with  A,  such  as  h  or  A  ,  the  value  of 
u  will  be  neither  a  maximum  nor  a  minimum  ;  but  if  the  power  o 

&  14 

h  be  such  as  will  not  change  with  /i,  as  It    or   A    ,  the  value  of 
will  be  a  maximum  when  that  term  is  essentially  negative,  and  9{ 
minimum  when  the  term  is  essentially  positive. 

75.  Finally,  it  may  occur  that  when  x  has  a  particular  value  a, 

Ihc  first  diffiirential  coefficient  -r-  will  become  infinite,  and,  therefore. 

ax  'J 


MAXIMA  AKD  MINIMA.  95 

m  order  t )  complete  the  search  for  maximum  and  minimum  values 

of  u.  we  ought  to  solve  the  equation  -r-  =  oo ,  and  if  a  be  a  root  of 

that  equation,  we  must  substitute  a  -f  ^h  and  a  —  A  for  a;  in  r^  =  Fx. 
Then  if  the  term  containing  the  lowest  power  of  h  be  found  to 
change  sign  with  h,  there  will  be  neither  maximum  nor  minimum ; 
but  if  not,  there  will  be  a  maximum  when  that  term  is  negative,  and 
a  minimum  when  it  is  positive. 

76.  Prop,  To  determine  the  maximum  and  minimum  values  of 
an  implicit  function  of  a  single  variable  x. 

Let  F  [x,  y)  =  0  be  the  relation  connecting  x  and  y, 

du 

Put  u  =  F{x^y)  =  (i',     ^hm    |  =  -  |. 

dy 

T^       1         .  .  .  .  dy     r.        du      ^   ^ 

But  when  y  is  a  maximum  or  minimum,  — -  =  0 ;  .•.—_  —  0  also, 

dx  dx 

and  we  have  the  two  following  conditions  by  which  to  determine  the 

values  of  x  and  y,  viz. : 

'£jl^^O...(l),     and     «  =  ^(.,y)  =  0. ...(.). 

Having  found  the  values  of  x  and  y  which  correspond  to  either  a 
maximum  or  minimum,  we  distinguish  one  from  the  other  by  sub- 
stituting the  same  values  in  the  successive  differential  coefficients, 
and  stopping  at  the  first  which  does  not  reduce  to  zero.  If  this  be 
negative,  y  will  be  a  maximum  ;  if  positive,  ia  minimum. 

The  successive  differential  coefficients  are  formed  Without  difficulty 

dv 
from  the  value  of  —-  already  found,  and  their  particular  values,  when 
(tx 

-—  =  0.  become  much  simplified. 

„,  du  du  d?u  d^u  „  .  , 

Thus,  put  -=;,,-  =  ff,  —  =p^,  —  =  ,„  &o.,  and  employ 

the  [  ]  to  represent  .the  particular  values  of  the  quantities  enclosed, 


96  DIFFEKENTIAL  CALCULUS. 

when  —  =  />^  =  0.     Then  observing  that  /?i  q-^  &c.,  are  usually  func- 
tions of  both  a:  and  y,  we  have 

tdjp^        d^    dy\_       (dq^   ,    dq^    di/\ 
^  _  _      W-g         <^y     ^-g/       ^^\dx  '^   dy'  dx) 

dx^  ~  q^ 


^1^ 


-0 


And  in  a  similar  manner  the  higher  differential  coefficients  can  be 
formed,  although  the  operation  is  more  laborious. 

77.  The  following  considerations  will  facilitate^  the  application  of 
the  preceding  principles  to  particular  examples  : 

1st.  If  a  quantity  which  is  a  maximum  or  minimum  contain  a 
constant  factor,  that  factor  may  be  omitted  and  the  result  will  still 
be  a  maximum  or  minimum. 

2d.  If  w  be  a  maximum  or  minimum,  then  w  ±  a  is  also  a  max- 
imum or  minimum,  but  a  —  u  will  be  a  minimum  when  w  is  a 
maximum,  and  a  maximum  when  w  is  a  minimum. 

3d.  If  tt  be  a  maximum,  -  will  be  a  minimum:    and  if  u  be  a 
u 

minimum,  -  win  be  a  maximum. 
u 

4th.  If  w  be  a  maximum  or  minimum  and  positive,  then  m^,  w^, 

ftnd  in  general  t^",  will  be  a  maximum  or  minimum  where  n  is  any 

positive  integer:    but  if  w  be  negative,  t*^,  w*,  and  in  general  u^^^ 

will  be  a  maximum  when  u  is  a  minimum ;  and  a  minimum  when  u 

is  a  maximum. 


MAXIMA  AND   MINIMA.  97 

5th.  If  w  be  a  nnaximum  or  minimum  and  positive,  log  u  will  also 
be  a  maximum  or  minimum. 

6th.  \i  the  power  u^"  be  a  maximum  or  minimum,  the  root  u  is 
not  necessarily  either  a  maximum  or  minimum ;  for  it  may  be 
imaginary ;  and  even  when  w^n  _  q  ^nd  a  maximum,  the  corw 
responding  root  i^  =  0,  although  real,  is  not  admissible  as  a 
maximum,  because  the  adjacent  values  of  u  are  imaginary. 

7th.  The  value  x  =  cc  cannot  correspond  to  a  maximum  or 
minimum  value  of  w,  because  x  cannot  have  a  preceding  and  a  suc- 
ceeding value ;  but  u  =  ao  may  be  a  maximum  provided  the  pro- 
ceding  and  succeeding  values  of  u  have  like  signs. 

8th.  In  determining  whether  u  is  a.  maximum  or  minimum  by 

the  sign  of  -^,  when  —  has  the  form  of  a  product  ?'i .  ^2  •  ^3 ^»» 

and  X  =:  a  causes  one  factor  v^  to  become  equal  to  zero,  the  only 

term  in  -r-^  necessary  to  be  examined  is  that  involving  — -^,  since 
dx^  ^    ax  , 

the  other  terms  disappear  with  v^. 

78.  1.  To  determine  the  values  of  the  variable  x  which  render 
the  function  u  =:  Qx  -\-  Sx"^  —  4x^  a  maximum  or  minimum,  and 
the  corresponding  values  of  the  function  u. 

Here       u  =  6x  ■}- Sx^  —  4x^.     .  • .  -^  =  6  +  6ir  -  I2x^  =  0. 

ax 

oil  13.,  1 

^-r=2       •••^  =  4^4=  +  ^     ''     -2 

Hence  if  u  have  maximum  or  minimum  values,  they  must  occur 

when  a?  =  1  or  when  a;  =  —  -• 

To   discover  whether  these  values  are   maxima  or  minima,  we 

form  the  second  differential  coefficient :  thus 

dPu 

--—  =  6  -  24a;  =  6  -  24  =  -  18       when,      x  =  1 

=  6  -H  12  =  +  18       when      xz=z  -L 
*•»  2 


98  DIFFERENTIAL    CALCULUS. 

. ' .  when  X  =  I,     m  =  6  +  3  —  4  =  5     a  maxiinum, 

1  3       17 

whan  X  =  —  -i     u  =  ^S  +  -  +  -=— -a  minimum. 

2  4       2  4 

2.  u  =  X*  —  8x^  -f  22ic2  —  24a:  +  12,  a  maximum  or  minimum. 

^  =  4x^  —  24a;2  +  44a;  -  24  =  0     or     x^  -  6x^  -\- Ux  —  Q  =  0, 
ax 

The  value  a;  =  1  is  obviously  a  root  of  this  equation,  and  by 

dividing  the  first   member  by  x  —  I   we  have  for   the  depressed 

equation 

x^  —  5x  +  6  =  0.     .-.x  =  2,     or     x  =  S. 

Hence  the  values  requiring  examination  are 

ic  =  1,     a;  =z  2,     and     x  =  3. 

But         -—■  =  12x^  —  48a;  +  44  =  4-  8     when     x  =  1, 
ux 

=  —  4     when     a;  =  2, 
=  +  8     when     x  =  B, 
,  when  X  =  l^'    u  =  S     a  minimum, 

when  a;  =:=  2,     u  z=z  4     a  maximum, 

when  a;  =  3,     u  z=S     a  minimum. 

3.  u  z=z  x^  —  5a;*  +  Sa;^  +  1     a  maximum  or  minimum 

du 

j-  —  b3^^  20a;3  +  15a;2  =  0      or      a;*  —  4a;3  +  3a;2  =  0  .  .  .    (1) 

.  • .  a:2  =  0,     or    a;^  —  4a;  4-  3  =  0, 

4ind  the  four  roots  of  (1)  are  0,  0,  1,  and  3. 

d^u  I  d^ij\ 

—  =  20a;3-60a;2+30a;=  0     when  x=0 ;  I .  • .  let  us  examine  — J- 

=  —  10  "    a;=l ;  then,  ?^  =  2,  a  max. 

=  +  90  "     a;=3 ;  then,  ti  =  ~  26,  a  min 

d^u 

—  =  60a;2  -  120a;  +  30  =  30     when     a;  =  0. 

,  • ,  f*  =  1  is  neither  a  maximum  nor  a  minimum. 


MAXIMA  AND  MINIMA.  99 

79.  In  each  of  the  preceding  examples,  the  condition  —  =  oo , 

renders    a;  =  oo ,  and  therefore  not  applicable  to  a  maximum  or 
minimum. 

Remark.  In  forming  the  second  differential  coefficient,  it  will  save 
labor  to  omit  any  positive  numerical  factor  common  to  every  term 
of  the  first  differential  coefficient,  and  the  sign  of  the  second  differ- 
ential coefficient  will  not  be  affected  by  such  omission. 

80,  Ex.  1.  ■M  =  -7 -f-  a  maximum  or  minimum. 

{x  +  'Zy 


du      3  (a;  4-  3)2(a:  +  2)  -  2  (.r  +  3)3      ^ 

-—  =  — ^^ ^—^ — — f-— ^ '—  =  0,      or,    -r-  =  00 


du 
dx~  (^  +  2)3  ~  "'      "''    dx 


But,  when  ^  =  0   we  have  Z{x  +  ZY{x  +  2)  -  2{x  +  3)^  rr:  0 

• .  re  +  3  =  0,  or,  3(a;  -f  2)  =  2(a;  -I-  3),  .  • .  x  =  -  Z,  or,  a:  =  0. 

d^u  _  6  (a;  +  3)  (a;  +  2)^-  12  {x  +  3)^(a;  +  2)  f  6  (a:  +  3)^ 
dx"^  ~  {x  +  ^Y 

9  27 

=  -    when    a;  =  0,  and  .  • .  u  z=:-—-  a  minimum. 
8  4 

=  0       "       a;  =  -  3. 

Now,  without  actually  forming  the  3d  differential  coefficient,  it 
is  easily  seen  that  it  will  contain  one  term  (and  only  one)  which 
will  not  reduce  to  zero  when  x  z=z  —  3 ;  and,  therefore,  the  corres- 
ponding value  of  u  is  neither  a  maximum  nor  minimum. 

27 
The  value    a;  =  0,     gives     u  =  —  &.  minimum, 

,.       ^  .      du       3(a:4-3)3(a;  +  2)-2(a:4-3)3 

Now  takmg  the  equation  —  =  — ^^ —j — -t-k^i — —  ~  oo , 

dx  \X  -f-  4i\ 

we  get  a;  +  2  =  0,     or,     a;  =  •—  2, 

and  by  putting  successively  a;  =  —  2  -f-  ^      and      a;  =  —  2  —  A, 
in  the  value  of  the  original  function  m,  there  results 


100  DIFFERENTIAL  CALC0LUS. 

''»  =  ^("  +  ^)  =  (-2+A  +  2)^-— A^— 
„,         .,        (-2-A  +  3)3       (1-A)» 

"' =  ^("  -  *)  =  (32irx+2p  =      A^— 

•nd   since   both  of  these   are   positive  values,  and    less   than  that 
orresponding  to  a:  =  —  2,  we  have  w  =  oo    a  maximum. 

2,  w  =  "7 — r— rf^i  a  maximum  or  mmimum. 
{x  +  1)3 

.'.a-  — 1=0,    or,     2(a;  + 1)  —  3(a?  — 1)  =  0,    or,    a; -f  1  =  0. 

.•.a:=l,     or,    a;  =  5,     or,     a;  =  —  1. 

^  _  2(a:  +  1)^  -12(a;  +  l)(.r  -  1)  +  12(a;  -  1)^ 
dx^  ~  (x-i-  1)5 

.  • .    -r-z  =  -  when  a:  =  1,  and  «  =  0,  a  mmimum. 
dx^       4:  ' 

=  —  — —  when  X  =  5,   and    u  = —-  a,  maximum. 
324  27 

When    a:  =  —  1,    w  =  oo ,    which  is  neither  a  maximum  nor  • 
minimum,  for 

but  «3  =  i?-(a-A)=(-^-f)<0, 

S    tt=:6-|-(^  —  a),a  maximum  or  minimum. 

-T-  =  rt  (^  ""  ^)  =  0,     ,  ' .  X  =  a,     and     «  =  6. 


MAXIMA   AND  MINIMA.  101 

Hence  wi  cannot  develop  by  Taylor's  Theorem.      Put    a  zt  hy 
for  X  in  the  value  of  u. 

and  «3  =  6  +  (a  -  A  ~  a)*=  6  +  (-  A)* 

This  last  value  is  imaginary,  and  therefore,  u  —  b  is  neither  • 
maximum  nor  a  minimum. 

4.     u  =  b  -\-  (x  —  a)  ,  a  maximum  or  minimum. 

—-  =  -  (a;  —  a)  =  0,     .  * .    ar  =  a,     and     u  =  b. 
ax       o 

cPu      4  'i- 

—- ^  =  -  (ar  —  a)     =  ao  ,  when  x  =  a.     Then  put  x  =  a  drA. 

.  • .   «2  =  6  +  (a  +  ^  -  a)^=  6  +  (+  A)^  >  b, 

End  ^3  =  6  +  (a  —  A  —  a)  =  6  +  ( —  A)    >  6,  also. 

.  • .  X  =  a  gives  w  =  6,  a  minimum. 

6.     M  =  6  —  (a  ~  a;)  ,  a  maximum  or  minimum. 
-— =  -(a  —  x)  =  0,      .  • .  a;  =  a,     and     w  =  5. 

-—=::——  (a  —  a;)   '=—00,  when  x  —  a.    Then  put  a;  =  a  ±  A. 

.',    W2=:6--(-A)*<6,     and     Wg  =  6  -  (+ A)*<  6,  also. 
.  • ,   X  =  a    gives     w  =  6,  a  maximum. 

I 
6.     u  :=  b  -^  {x  —  a)  +  c  {x  —  a)2,  a  maximum  or  minimum, 

|  =  |(.-a)*+2c,(.-«)  =  0. 

5  4 

•      T  —  rt  =  0,      or,     -  +  2c(^  —  a)^  =  0. 


m 


DIFFERENTIAL    CALCULUS. 


125  3125 

'.  x=:a,&nd  u  =  b,  or,  x=a-  --—,  and  u  =  b  - -— 


216c3' 

d^u       10 ,  v-i  .   ^ 

But     -z-  =  —  (a;  —  a)     +  2c  =  oo  ,      when  x  =  a 
oLx        y 


46656c* 


-c>0when^  =  a-^^3 


Hence  when    x=.  a  — 


125 


>     we  have    u  =:b 


3125 


216c3  46656c5 

a  minimum.    , 

In   order   to  examine  the  value  of  a;  =  a,  put   a  zfc  A  for   x   in 
the  original  value  of  u. 

.  •.  2^2  =.  6  +(+  A)*+  c(-h  A)2  >  5,  2*3  =  6  +(-  /^)*  +r(-A)2<6. 

.  • .  t<  rz:  6  is  neither  a  maximum  nor  a  minimum. 

7.  To  inscribe  the  greatest  rectangle  in  a 
given  circle. 

Put  the  diameter  AC  =.  a,  arid  the  side 
AB  =  X  ;  then 

ADz=z-y/a^-x^   and   ^^X^i)=a;v/a2-a.'2. 
.  • ,  -M  =  {AB  X  AJ)y  =  a;2(a2  —  x'^)='d  max. 


=  2a22;  —  4x^  =  0.     .  • .  ar  =  0,     or     x  =  a 


-—  =  2a2  —  12a:2  _  2a2     when     a;  =  0 
rfa;2 


4a2 


X  =z  a 


AD  =  \/  a2  -—  -  a2  =  a  \/  - 


AB, 


.    1 


and  the  rectangle  must  be  a  square.     Its  area  is  -  a^. 

8.  To  inscribe  a  maximum  cylinder  in  a  given  right  cone  having 
a  circular  base. 


MAXIMA  AND  MINIMA. 


lOS 


Put  JO  the  radius  of  the  cone's 
base  =  6,  CO  the  altitude  of  the 
cone  =  a,  DF  the  altitude  of  the 
cylinder  =  x. 

Then  from  the  similar  triangles 
CO  A  and  CQD  we  have 

CQxOA 


CO'.CQ-r.  OA:QD  = 


CO 

(a  —  x)b 


n/- 9. ^ 

F  /"'^      0  ^N& 


• .  volume  of  cylinder  =  — ~ 


.  • .  M  =  (a  —  x^x  =  a^x  —  2ax'^  -f-  ^^  =  maximum. 

,  ~z-  =  a^  —  4ax  +  Sx^  =  0,     or     x'^  —  -ax  ==  —  rr a'. 
dx  '33 

2     ^1  1 

X  =  -a±-a  =  a     or     =-a. 

t>  O  O 

d'^u 

-r-^  —  —  4a  •\-  Qx  =z  2a      when     a;  =  a 


2a 


1 


Hence  the  altitude  of  the  cylinder  is  one-third  of  the  cone,  and  con- 
sequently 

4  4 

volume  of  cylinder  =  —  'rtab'^  =  -  volume  of  cone. 
^1  y 


9.  Find   the   greatest   and   least  ordinates  of  the  curve  whose 
equation  is  a^i/  —  ax^  -f-  a;^  =  0. 
Put  u  =  a"^!/  —  ax^  +  x^  z=  0  ,  ,  ,  .   (1). 

du 


Then 


dx 


—  —2ax-\-  3a:2 


(2). 


Combining  (1)  and  (2),  we  get 


2  4 

a:  =  0     and     y  =  0,     or     x  — -a     and     y  =  —-a. 

O  ,61 


104  DIFFERENTIAL  CALCULUS. 


But  ^  =  a«,     ^=.-2a-U6^, 

Ldx'A  Ldx'^A       VdyA       a 


2     ,.  2 


a;  =  -a. 


2  4 

• .  When  a:  =  0,  y  =  0,  a  min.,  and  when  a;  =  -«,  y  =  — a  —  a  max. 


10.  To  find  a  number  re  such  that  its  x*^  root  shall  be  a  maximum. 

du 
dx 


w  =  ar*  =  a  maximum.     —  =  ic*     (1  —  log  a;)  =  0. 


.1_2 

.  • .  a?  *       =0     or     1  —  log  a:  =  0. 

The  first  of  these  equations  gives  a:  =  0 ;  the  second  log  a*  =  1 ; 
1 
whence  x  =  e  and  m  =  e«  =  maximum. 

In  this  and  in  many  similar  examples,  we  may  draw  the  final  in- 
ference without  forming  the  second  differential  coefficient,  it  being 
obvious  from  the  nature  of  the 
question  that  there  is  one,  and 
only  one  maximum,  and  it  be- 
ing easy  to  decide  which  of  the 
values  of  x  is  that  applicable 
to  the  maximum. 

11.  To  cut  the  greatest  para- 
bola from  a  given  right  cone 
having  a  circular  base. 

Put  AB  the  diameter  of  the  ^' 
base  =  a^  AC  the  slant  height 
=:  6,  and  BG  =  x. 


Then  ^  G"  =  a  —  ar,  and  by  the  property  of  the  circle, 


•  FE  =  ^FG  =  2y/x{a-x] 


MAXIMA  AND  MINIMA.  105 

Also  by  the  similar  triangles  BA  C  and  B  OD^  we  have  /- 

BA..AC:..BG..   GB  ^^^^  J^  '  ^y     ^  ^ /. 
But  tnc  area  of  the  parabola  •      'V    ^^^  l         ^  /^         *^ 

o  A  bx   / ^/  *        y^ 

FDE  z=^FEx  GD:=-'  ~Jax  -  x\  V>  A, 

3  3     a  ^  fjy  /' 

T  3  <  ^  I   V  ^ 

.•.  ■w=aa;3— a;*=  max. :    -— ii=3aa;2— 4a;3=0,  and   a:=0  or  x=-a,  ^\  / 

^    dx  4   '  ^1 


the  second  value  being  obviously  that  required,  since  when  a;  =  0 
the  area  of  the  parabola  =  0. 

.  •.  area  of  maximum  parabola  =  -ab-y/^. 

12.  To  form  the  greatest  quadrilateral  with 
four  given  lines  taken  in  a  given  order. 

Put    AB^a,   BC=h,    CD  =  c,   DA  =  e,     A^ 
angle  BAD  =  x,  and  BCD  =  ar^,  the  latter  an- 
gle Xj  being  obviously  a  function  of  ar,  since  the 
two  are  connected  by  the  relation 

[BDy  =  a2  4-  e2  _  2aecosx  =  b^  +  c^  —  26c.cosa;i (I). 

But  area       ABCD  =  AABD  +  aBCD  =  -  ae  sin  a;  +  ]-  be  sin  x^. 

.*.  «=acsina;4-^<^.sinari=:amax.,  and  -—  =aecosa;-|-6ccosa:,  —  =0. 

ax  ^dx 

Now  by  differentiating  (1),  we  have 

.         dx^ 
ae .  smx  =  be .  sni  x-,  — r-^» 
dx 

dx-,       ae .  sin  ar  du  ,    ,  ae  sin  x 

,'  .-r-  z=z  - — .     .'.—-  z=ae.  cos  X  -\-  be.  cos  x-,  - — : =  0. 

ax        be .  sm  x^  dx  be  sm  x^ 

,  • .  sin  2  cos  x^  -\-  sin  x-^  cos  x  =  0,     or     sin  {x  +  x^)  =  0. 

.  • .  a;  4-  a?i  =  0,     or     a:  -f  ar^  =  180°. 

The  latter  is  plainly  the  required  solution,  and  consequently  the 

quadrilateral  must  be  such  as  can  be  inscribed  in  a  circle. 


106  DIFFERENTIAL  CALCULUS. 

13.  To  find  the  greatest  quadrilateral  that  can  be  contained  within 
a  given  perimeter. 

Suppose  ABCB  to  be  the  required 
figure,  and  suppose  two  of  the  sides  x  and  y 
to  vary,  while  the  other  two  sides  v  and  z 
and  the  diagonal  t  remain  unchanged. 
Then,  since  ABCB  is  supposed  to  be  the 
greatest  quadrilateral  which  can  be  formed 
with  the  given  perimeter,  the  triangle  ABC 
must  be  greater  than  any  other  triangle  having  the  same  base  /,  and 
the  sum  of  the  sides  z=z  x  -{■  y  z=z  b  ?i  constant. 

But  if  X  -{■  y  -^  t  =  s, 

the  area  oi  i\iQ  AABC —\/ -si- s  —  x\l-s  —  y\l-s  —  t\ 
Therefore,  by  squaring  and  omitting  the  constant  factors 
-  s     and     -s  —  t,     we  have 

u  —  iry^  —  ^)  \^^~y)  ~  (o*  ~  ^)  (^'^  ~  ^  +^)  =  ^  maximum. 
=  I  -  s— a:)  —  r-  s—b-\-x\z=zO^  or  6— 2j=:0,  and  .  • .  xz=:-b. 


du 
dx 


,'.    y=zb  —  x=:x  =  -b. 
til 

that  is,  the  sides  AB  and  BC  must  be  equal.  Similarly  it  may  be 
shown  that  x  —  v^  v  =  z^  z  =  y. 

Hence  the  figure  must  be  equilateral,  and,  consequently,  either  a 
rhombus  or  square.  But,  since  the  lengths  of  the  sides  are  now 
given,  the  quadrilateral  must  admit  of  being  inscribed  in  a  circle. 

.  • ,  The  figure  must  be  a  square. 

14.  To  find  the  greatest  figure  of  n  sides  contained  within  a  given 
perimeter. 


MAXIMA  AND   MINIMA.  107 

,  By  supposing  two  sides  AB  and  BC  to 
vary,  while  the  other  sides  remain  fixed  in 
magnitude  and  position,  we  prove,  as  in 
the  last  example,  that  AB  =:  BC,  and 
similarly  that  BC=  CD,  CD  =  DE,  &c. 
Therefore  the  required  figure  must  be 
equilateral.  , 

Then,  supposing  the  three  equal  sides  HA,  AB,  and  BC,  to  vary 
in  position,  while  the  other  sides  remain  fixed,  we  show,  as  in  a  pre- 
ceding example,  that  the  circumference  of  a  circle  can  be  described 
through  H,  A,  B,  and  (7;  and,  similarly,  that  a  circumference  can 
be  drawn  through  A,  B,  C,  and  D.  But  only  one  circumference  can 
be  drawn  through  the  same  three  points  A,  B,  and  C.  Therefore 
the  same  circumference  passes  through  H,  A,  B,  C,  and  D.  And, 
similarly,  it  may  be  shown  that  this  circumference  passes  through 
JS,  F,  G,  &c.  .  * .  The  polygon  must  be  equiangular,  and,  conse- 
quently, regular. 

15.  To  divide  a  line  a  into  n  parts,  x,  x-^,  a:,,  &c.,  and  determine 
the  relations  between  those  parts  when  the  continued  product  of 
their  numerical  values  shall  be  a  maximum. 

Let  two  of  these  parts  x  and  x-^  vary,  while  x^^  a-g,  &c.,"remaiH 
constant. 

Put     iCg  +  ^3  +  <fcc.  =  b,  and  X2X  x^x  x^  &c.  =  c 

Then  x  -\-  x-^^=ia—h,  and  xx^'X^'X^  &c.  =  x{a  —  b  —  x)  c. 

.*.  u=:x{a  —  b  —  x)-=z£k  maximum,  -—  =  a  —  6  —  2a;  =  0 
^  '  ax 

.  • .  xz=-{a  —  b),  and  x-^  =  a  —  b  —  x  =1  -{a  —  b)  z=  x. 

Similarly,  X2  =  x,  x^z=  x,  (fee,  and,  therefore,  the  parts  are  all 
equal. 

10.  To  determine  the  number  of  equal  parts  into  which  a  given 
number  a  must  be  divided,  so  that  their  continued  product  may  be  a 
maximum. 


108  DIFFERENTIAL   CALCULU^ 

Let  X  •=.  required  number  of  parts ;  then  -  =  value  of  one  part. 


a       a      a    ^  ^  /aY 

-  X  -  X  -   &;c.  to  X  factors  =  I  -  J  ; 


a  maximum. 


X  X 


«=log  I     j  =  a:  (log  a— log  a;)  =:  a  max.  — =:loga— logar— 1=0 


. ' .  U)^  -  =  I.    -  =  e,     and     x  =  — 
'XX  e 


This  is  a  solution  in  the  arithmetical  sense  only  when  a  is  a  mul- 
tiple off,  {' >v  otherwise  x  would  not  be  an  integer. 

The  general  solution  belongs  to  the  following  problem.     To  find 


th 


a  number  x  such  that  the  x     power  of  -  shall  be  a  maximum. 

17.  To  determine  the  point  P,  in  the  line  joining  the  centres 
C  and  C\  of  two  unequal  spheres,  from  which  the  greatest  amount 
of  spherical  surface  can  be  seen. 


PutC'0  =  r,   CiOi  =  ri,   CCy^  =  a  CP  —  x,   C^P  z=z  x^  =  a  ^  x. 


,'.   CD  =  —,  aZ),  =li-,  DEz=zr 

X  ^      ^  X.  X 


r^        r{x  —  r) 


ftnd  similarly 


A^i  = 


r,{x^  -  ^i) 


*.  Surface  of  zone  OEQ  =  2  rrr 


r(x  —  r) 


OyEj^Qi  =  2';rri 


'•iC^i-^) 


MAXIMA  AND  MINIMA.  109 


^3  J.  3 

^2  _^  ^  2 1 —  --  max. 

X       a  —  X 

J  r* 


u 

r^x-^ 

"■'l 

T^H^-T^  _ 

X 

T^ 

^1 

du 

r3 

r,3       _ 

{a^xf- 

J 

,  and  surface  seen 

Kr^ 

which  is  always  less  than  the  entire  surface  of  the  two  spheres. 

18.  A  right  prism,  whose  base  is  a  regular  hexagon,  is  truncated 
6y  three  planes  drawn  through  the  alternate  vertices  of  the  upper 
base,  and  intersecting  at  a  common  point  in  the  axis  prolonged. 
Required  the  inclinations  of  the  planes  to  the  axis,  when  the  truncate 
prism  shall  (with  a  given  volume)  be  contained  under  the  least  surface. 

Let  ABC  DBF  be  the  lower  base  of  the  prism,  and  abcdf^  the 
upper  base. 

Join  /6,  bd^  and  df^  and  through  these  lines  draw  planes  inter 
secting  the  axis  Rr  prolonged  at  some  point  v. 

The  plane  fvb  intersects  the  edge  Aa  at  aj,  cutting  off  from  the 
prism,  the  triangular  pyramid  fbaa-^.  From  r,  the  centre  of  the 
upper  base,  draw  rf,  ra,  and  rb.  •  Then  fabr  is  a  rhombus,  whose 
diagonals  bisect  each  other  at  o  perpendicularly.  Join  va-^ ;  it  will 
be  perpendicular  to  /6,  and  will  pass  through  o.  Then  ao  =  or^ 
and     .  • .  aa^  =  rv. 

. ' .  Pyramid  fbaa^^  is  equal  to  the  pyramid  fbrv, 

. ' .  The  volume  of  the  prism,  when  terminated  by  the  three  planes 
which  intersect  at  v,  is  equal  to  the  volume  of  the  original  prism, 
for  all  inclinations  of  the  planes. 

Put  Bb  =  a,     AB  =  b,     the  angle  rvo  =  x. 

Then    ro  z=z  oa  =  -b,   aa^z=:  -b  cot  ar,    oa^z^ov  =-b  cosec  a?, 


o/=z  ob  =  -b -/3.     Aa^  ~  a  —  - 6 cot ar. 


no 


DIFFERENTIAL  CALCULUS. 


.  * .  Surface  ABbai=- h  (2a  —  -h.coix). 
lit  li) 

Surface  a^bvf  =  fb  x  vo  =  -b^  y/Z  coseca;. 

Hence  by  the  nature  of  the  question,  we  shall  have 

1  3         - 

QABba^  4-  Sa^bi>f=  36  (2a  —  -  6  cot  rr)  -\-  ^b^  -/S  cosecx  =  a  min. 

.  • .   u  =2a  ~  ~b  cot  X  +  -b  y^  cosec  a;  =  a  minimum. 

-J-  =  -  6(cosec2a;--  WS  cosec  a;  cot  a;)  =  0.       .  • .  cos  x  =  — - 
dx      2   ^  ^  ^  ^3 

.  • .  a;  =  54°  44'  08". 

This  is  the  celebrated  problem  relating  to  the  form  of  the  cellt 

of  the  bee. 


CHAPTER    IX. 

FTIKCT20^S    OP    TWO    INDEPENDENT    VARIABLES. 

81.  Hitherto  it  has  "been  supposed  that  the  function  u  depended, 
either  directly  or  indirectly,  on  a  single  variable  x.     But  the  value 

,of  u  may  depend  on  the  values  of  two  or  more  variables,  entirely 
independent  of  each  other.     Thus,  if  there  were  given 

u  =  xy  -\-  2/2,  ....  (1). 

we  might  suppose  x  to  vary  and  y  to  be  constant;  or  y  to  be 
variable  and  x  constant ;  or,  lastly,  x  and  y  may  vary  simulta- 
neously.  These  three  suppositions  lead  to  three  essentially  different 
changes  in  the  function  w. 

Thus  when   x  becomes    x  -\-  h,  and   y   is   constant,  u   becomes 
Uy  —  xy  -{■  hy  '\-  y^. 

When  y  becomes    y  -\-  k^  and   x  is   constant,  u  becomes 
^2  =  xy  4-  xk  ■\-  y"^  -{-  2yk  4-  Jc^. 

And  finally,  when  x  and  y  become  respectively  x  -{■  h  and  y  +  ^, 
u  becomes     u^  =  xy  -{-  hy  -{-  kx  -\-  y"^  -\-  2ky  -\-  k"^  -\-  hk. 

The  general  case  is  presented  in  the  following  proposition. 

82.  Prop.  Having  given  u  =  F{x,  «/)....(!).,  to  develop 
v^  z=  F  [x  -\-  h  y  -{-  k),  the  variables  x  and  y  being  independent 
of  each  other. 

Since  x  and  y  are  supposed  to  have  no  mutual  dependence,  they 
may  be  supposed  to  vary  successively. 

Let  X  take  an  increment  h ;  then  u  becomes  u^  =  F  (x  -^  h,  y) 
which,  developed  as  a  function  of  x  -i-  k  by  Taylor's  Theorem,  gives 


112  DIFFEREN-TIAL  CALCULUS. 

''•  =  "+^-i+^^-t:2+^-3-i:2:3+*'^  (2)- 

m  which  w,      — ,     -7-2 ,     &c.,  are  functions  of  both  x  and  y. 

Now,  if  in  every  term  of  (2),  we  replace  y  by  y  -f  k,  we  shall 
convert  u^^  =  F{x  -\-  h,  y)  into  u^  z=z  F (x  +  h^  y  -\-  k)^  and,  since 
each  term  in  the  second  member  of  (2)  will  then  be  a  function 
of   y  -\-  k^  we  must  replace 

-  du   k      d^u      k^ 

u   by    „+-.-  +  _._  +  &0. 

ydu  du 

du   .        du  ^       dx    k  dx     k"^         . 

^  ^^   ^  +  ^-T  +  -^-TT2+^^- 

d^u  dhi   ,      dx"^    k  dx^      P 

dx^    ^^    (/^2+-^-l+-^T72  +  &c- 

dH  d^u 

d^u  d^u   ,       dx^    k  dx^      k"^         . 

rf?    ''y    1^5  + -^  •  1  +  ^^  •  O  +  *^"- 

c/j7        d^u 
But  we  put  for  convenience   —-z — =  ,   indicating   thereby 

that  two  differentiations  of  w  have  been  performed,  the  first  with 
respect  to  ar,  and  the  second  with  respect  to  y.     Similarly  we  put 

^du  ,  d'^u 

d^  —  d  — 

dx         d^u  djp"         d?u 

-;^  =  Srf^'  ""''  ^r  =  rfi^'  *'  '^'''  expression  indi- 
eating  one  differentiation  with  respect  to  rr,  followed  by  two  with 
respect  to  y;  and  the  second  implying  two  differentiations  with 
regard  to  a;,  followed  by  one  with  regard  to  y.  And  generally,  we 
denote  the  result  of  n  differentiations  with  respect  to  a;,  followed  by 
w  differentiations  with  respect  to  y,  by  the  symbol. 

dx*dy^ 


FUNCTIONS  OF  TWO   INDEPENDENT  VARIABLES.  113 

Now  let  the  necessary  substitutions  be  made  in  (2),  and  we 
shall  get 

dii   h       du    k       d'^u      h?"  dhi      hk   ,d^u      k^ 

^^'^'^'^d^'i'^d^'l'^  H^'Tyz^dld^  'T'^df'T^ 

d^u         A3  ^Zy^      h2jc       dH     hk"^      dhi       k^ 

'^d'x^'  I  .2.S~^  dchl^'Y^'^  dxdt/^'U2'^df^'  l.2,S  "*"     ^* 

which  is  the  proposed  expansion. 

If  we  had  supposed  the  variable  y  to  receive  its  increment  first, 

we  should  have  obtained  the  following  series  for  u^. 

du    k       du    h       d^u      k"^         d^u      kh   *   d^u      h^ 

**  ~^'^d^'i'^di"i'^d^  '  T72  "^  di/dx  'T'^d^'T^ 

dhi        k^  dH      k'^h         d^u      kh?       dhi       h'^ 

^  If  '  1.2.3  "^  dfdx  '  172  "^  di/dx^  '  172 /"  ^  *  1.2.3         ^* 

The  two  series  must  obviously  give  equal  results,  and  being  true 
for  all  values  of  k  and  k,  the  coefficients  of  the  like  powers  and  pro- 
ducts  of  h  and  k  must  be  equal. 

d'^n         d^it  d^u  d^u  d^u  d^u 


&c. 


dxdy       dydx      dx'^dy        dydx^       dxdy"^        dy'^dx 
Hence  the  result  of  n  differentiations  with  respect  to  t,  followed 
by  m  differentiations  with  respect  to  y,  will  be  the  same  as  that  pro 
duced  by  performing  the  differentiations  in  a  contrary  order. 

EXAMPLES    OF    DIFFERENTIAL    COEFFICIENTS. 

83.  1.  w  =  xhj  -f-  a2/2. 

du        „  „        du         o       ^  d'^u        ^  d^u        ^ 

d^u  dhi        „■       . 

3x^    and     - — -  =  oz^  also. 


dxdy  dydx 

iPu  d^H  dhi    _  d^u  d^u  d^u 

d^  "^    ^'    df"^'     1^  =  6x  =  j^^     --j^  =  0  =J^^ 

dx*  ~-  ^'        dxUy  ~  ^  "^  dydx^'     dxHy-^  "  dy-'dx^'       *'     ^ 

8 


114  DIFFERENTIAL   CALCULUS. 


,  X      du  y  du 


y      dz       x^  +  y^      dy  x^  -\-  y^ 

cPu  x"^  —  y"^         dhi 


dxdy       (x^  +  y^y       dydx 


(fee,  &;c. 


„  .  da  du  .  , 

3.     u  =:  smx .  cos  y.     -7-  =  cos  x .  cos  y-r-=  —  sin  a; .  sin  y. 
^      dx  ^'dy  ^ 

dhi  .  d'^u       d^u 

-    ,    =  —  cos  a; .  sin  y  =  ,    ,  ?     — — -  =  —  sin  a;  cos  y 
dxdy  dydx      dx^ 

dhi  .         .  d^u,  dhi 

=.  sin  ic  sin  y  =  =  &c. 


dx'^dy  dydx^       dxdydx 

In  general  the  order  of  the  differentiations  is  immaterial,  provided 
we  always  differentiate  the  same  number  of  times  with  respect  to 
the  same  variable. 

The    expressions    —  and  —   are  called  partial  differential  co- 

du  du 

efficients:   -T~dx    and    -^^y   are    called  partial   differentials,  and 

du  z=. -T- dx  •\-  -T-  dy  is  the  total  differential  of  u. 
dx  dy 

84.  Similarly,  if  m  =  F{x^  y,  2),  where   ar,  y,  and  ^,  are '  inde- 

pendent  variables,  then 

du  .         du  .        du  . 
du  =:  -—  dx  -\-  -r-  dy  -\-  -r-  dz. 
dx  dy  dz 

And  generally,  to  differentiate  a  function  of  several  independent 

variables,  we  must  differentiate  successively  with  respect  to  each, 

and  add  the  results. 

85.  If  it  were  proposed  to  develop  Wj  =  F[x  -{■  h^  y  -{-  Jc^  z  -{■  l)^ 
where  u  =  F{x,  y,  z),  we  should  obtain,  by  supposing  ar,  y,  and  z 
to  vary,  and  reasoning  as  in  the  expansion  of  F(x  -\-  h,  y  -\-  k)^ 

du  h      du    k      du    I      dhi     h^  d^u     hk      d?\i     k^ 

^^'''''^di'\'^d^'l~^d^'l^dx^'T72'^d^'T'^d^/'r^ 
d^u      M_,^J^       dh(^    kl       dhi        P 


dxdz      1     '    dz^    1.2       dydz     1     '   dx^    1.2.3 
d^u       hH'         d^u       h^(i^       P  ^Ihi      hH 

dx-'dy  '  1.2"^^?^]^'l.2'^t/y3'r273"^^i2^*IT2'^ 


FUNCTIONS  OF  TWO   INDEPENDENT  VARIABLES.         116 

Remark.    The   formula    du  =  -;- dx  -{-  -r  dy  -\-  —- dz  -\-  &c.,  for 

dx  dy  dz 

differentiating  a  function  of  several  variables,  may  be  deduced  im- 
mediately fVom  the  preceding  development. 

For  put  k  =:  rh^  I  =  rji^  &c.  where  r,  r^  &c.  are  arbitrary,  since 
ar,  y,  z,  &;c.,  are  independent  of  eaoh  other.  Then  by  substitution 
and  reduction, 

U^    —     U  du  dll  «^W      .  .  ^  •  T  TO         B 

— i— =  -r-\-r-r--\-r.—r-  &c.  +  terms  m  h.  A^  &c. 

h  dx  dy  dz 

and  by  passing  to  the  limit,  making  h  =  0,  neglecting  terms  con 

taining  A,  A*,  &;c.,  and  finally  making 

M,  —  u  =  c?w,     h  =  dx,     rh  :=  k  =  dy,     r->h  =:  I  =z  dz,  &c.,  we  get 

du  .     .   du  .     ,   du  .     ,    „ 
du  = -r- dx  -{-  -T  dy  +  —- dz  -{-  &c 

dx  dy  dz  ^^^y>L 

86.  Prop.  To  differentiate  successively  u  =z  F(x  ,y). 
We  have  already  found  the  first  differential 

du  du 

du  =z -— dx  -^  -r-  dy. 
dx  dy 

Differentiating  this  and  observing  that  — ,  and  —   are  usuallj 

functions  of  both  x  and  y,  but  that  dx  and  dy  are  constant,  we  get 

„         d^u   ,  „         d'^u      ,    -     ,     d'^u     ,    _     ,   d^u     .  „ 

and  by  differentiating  again,  we  have 


116  DIFFERENTIAL  CALCULUS. 

and  similarly  may  i^w,  g?^w,  &;c.,  be  found,  the  numerical  coefficients  of 
the  several  terms  proving  the  same  as  in  the  powers  of  the  binomial. 

Implicit  Functions  of  two  Independent  Variables. 

87.  Pvo^.  Let  F{x^y,  s)=  0,  so  that  z  shall  be  an  implicit  function 
of  the  two  independent  variables  x  and  y,  and  let  it  be  proposed  to 
form  expressions  dz^  d'^z^  dec,  without  solving  the  equation  with 
respect  to  z. 

Put  u  z:z  F (x^y^z)  =:  0  \  then,  observing  that  u  is  directly  a  func- 
ion  of  the  independent  variables  x  and  y,  and  also  indirectly  a  func- 
;ion  of  x  and  y  through  0,  we  shall   have  for  the  total  differential 

.oefficientg]     and     g] 

id^rX       du    dz       du       ^      ,,.         ^  \d^i~\       du   dz      du     ^      ,^. 

du  du 

dz  dx       dz  dy 


dx  du^     dy  du 

dz  dz 


du  du 


dz  dz  dx  dy 

,  ' .  dz  :=z  —- dx  -\-  ~j-  '  dy  z=z —dx -rdy. 

ax  dy  du  du 

dz  dz 

Next  to  form  d^z^  we  have 

But  by  differentiating  (1)  with  respect  to  x,  (2)  with  respect  to  y, 

and  (1)  or  (2)  with  regard  to  y  or  x,  respectively,  and  observing  that 

du  du  du  .       .  ^  , 

-7-,  -r-,  ~r-,  arc  functions  01  x,  y,  and  z,  we  get 

dx  dy    dz 

dhi     dz_       d?u    dz^       du   dH       d^u  _ 
d^'di'^'d^''d7''^dz'dx''  ^ dx''  " 


2 


ELIMINATION   BY  DIFFERENTIATION.  117 

d^u      dz       d^u,    dz^       du    d?z    .    d^u 


dydz    dy       dz^    dy"^       dz    dy"^       dy"^ 


d^u     dz       d?u    dz    dz       du     d^z  d^u  d'^u      ^^  __  f. 


<j?yrf2;   dx       dz^   dy   dx       dz  dxdy        dxdy        dxdz     dy 

d-z    d'^z  d^z 

whence  — ^,  — — ,  and     ,    ,     may  be  found  in  terms  of  the  partial 
dx^    dy^  dxdy         *' 

differential   coefficients  of  the  first   and   second   orders  of  w,  with 

respect  to  x,  y,  and  2,  all  of  which  are  easily  formed. 

88.  Prop.  Having  given  u  =  92,  and  z  =  F{x,y),  to  differentiate 
u  without  previously  eliminating  z. 

If  we  suppose  x  alone  to  increase,  it  will  impart  a  change  to  « 
through  z ;  and  a  similar  change  will  be  transmitted  to  m,  when  | 
alone  varies  ;  thus  we  shall  have 


du 

du 

dz 

and 

du 

du 

dz 

dx' 

~  dz 

dx 

dy- 

dz 

dy 

.         du     .        du  ,         du    dz  ,      ,    du    dz     . 

du  z= dx  -{-  -r-  dy  =  —r  '  -r  dx  -{ — t- '  dy, 

dx  dy  dz    dx  dz   dy 


Elimination  hy  Differentiation, 

89.  When  a  constant  is  connected  with  a  function  by  the  sign 
-f-  or  — ,  it  disappears  by  differentiation  ;  but  when  it  is  a  coeff.cient 
of  the  function,  it  will  appear  in  the  differential  also. 

Thus,  if  u  =  F{x,  y)  =  0  ...  (1)  be  a  relation  connecting  x  and  y, 
into  which  the  constant  a  enters  as  a  factor,  then  a  will  also  be  found 
n  the  equation. 

rdu"}       du       du    dy        _  /         di/\ 

Now  a  may  be  eliminated  between  (1)  and  (2),  and  the  resulting 

dv 
equation,  called  a  differential  equation^  will  contain  x,  y,  and  ■—' 

dx 


li$  DIFFERENTIAL    CALCULUS. 

If  it  were  required  to  eliminate  two  con'stants,  we  might  differen- 
tiate twice,  thus  obtaining  three  equations,  including  the  primitive, 
(1),  with  which  the  elimination  could  be  effected,  and  the  resulting 

equation  would  contain  x,  y,  — ,  and  -r-^-     Surds  and  transcendental 

quantities  may  also  be  eliminated  by  a  similar  process. 

90.   1.  Given  y^  =  2ax,  oy  u  z=  y^  —  2ax  =  0,  to  eliminate  2a. 

tdiil       ^    dv       ^         ^  ^         V^      ^    dy 

3,-2.1  =  0, 

an   equation  in  which  2a  does  not  appear,  but  which  implies  tlie 

same  relation  between  x  and  y. 

I 

2.  Eliminate  the  surd  from  the  equation  y  =  (a^  -f  x^) (1). 

3.  Eliminate  a  and  b  from  the  equation  y  =  ax^  -\-  bx . . . ,  (\), 

|  =  2«.  +  6...(2)g  =  2a (3). 


By  combining  (1),  (2),  and  (3). 

/di 


2  dx^^     \dx  dx^ 


or  i!^^?.f^  +  ?y=0. 

dz^       x' dx       x^ 

I,  Eliminate  the  exponential  from  the  equation  y  =  2ae^', 

dy      ^ 

~-=  2ace^'  =  cy, 

dx 


ELIMINATION   BY   DIFFERENTIATION.  119 

5.  Eliminate  a  and  h  from  the  equation  y  =  a  cos  2a;  +  ^  sin  2flJ. 
~=  --2a  sin  2x-\-2b  cos  2x,  — -^  =  —  4a  cos 2a;  —  46  sin  2a:  =  —  4y* 

•••S  +  ^-o- 

91.  Prop.  Let  tt  =  7^0,  and  z  =  cp{x,  y),  where  a;  and  y  are  in- 
dependent variables,  and  let  it  be  proposed  to  eliminate  the  func- 
tion F. 

Differentiate  u  first  with  respect  to  ar,  and  then  with  respect  to  y, 

du       du    dz       dFz  d(p{x^  y)  .  . 

'    '    dx       dz     dx        dz         dx       '  '  '  '  \  i* 

du      du    dz       dFz  d(p{x,  y)  . 

dy~dz     dy  ~    dz  dy       '  '  '  '  \  r 

Now  divide  (1)  by  (2),  observing  that  the  common  factor  —r- 

will  disappear; 

du       dcp  (ar,  y) 

dx  __       dx  du    d(p(x,  y)      du    d(p(x,  y) 

'    '  du^      d(p  (ar,  y)  ^      dx         dy     ~  dy         dx 

dy  dy 

in  which  equation  F  does  not  appear. 

1.  Eliminate  the  function  F  from  the  equation  u  —  F{ax^  -\-  by^). 

Put     ax^  +  by^  =  z.     ,-.  ^=:  Sax'',      and       ~=z  2hy. 

dx  dy 

du    ^,         du   ^     ^ 
.-.  -—'2by  =  -—'Zax\ 
dx  dy 


2.  Eliminate  the  function  F  from  the  equation  u  =  -^(-) 
dx~        a;2      \y)      X    Ay)' y      ^^^       dy  ~       x     Ay) y' 
^    '    Idx'^  xJ^L-y^J~       xy^     \y)-  dy^  xy 


du   ,      du 
'.  u-^x- — \-y--  =  0. 
dx         dy 


120  differp:ntial  calculus. 

3.  Prove  that  if  y  =:  as'mx  -^  b  sin  2^,  then 

dx*  dx^ 

y  =  a  sin  ar  -f  ^  sin  2a;.  .  .  .  (1 ).       -^  =  a  cos  ^  -f  26  cos  2ar. 

— ^=:— Of  sina:— 4/>»sin2;j?.  . .  .(2).  and  — ^=asina:-f  16isin2a:. .  (3) 
dx^  dx'^  ^  ' 

Multiply   (1)  by  4,  and   (2)  by  5,  and  add  the  results  to  (3); 
thus 

92.  Proj).  To  determine  whether  any  proposed  combination  of 
X  and  y,  as  F{x^  y)^  is  a  function  of  some  other  combination,  as 

Put    u  =  F(x,  7/),     and     z  =  (p(x,  y);  then  if  i*  be  a   functicn 
of  z,  we  must  have 

du       du    dz  J      ^^  _  ^^^    ^^ 

dx^  dz    dx  dy~  dz    dy 

.  du    dz       du    dz 

'  dx    dy  ~  dy    dx 

which  is  the  required  test. 

1.  Is  u  =  x^—  6x^y+  V2xy^—  8y^  a  function  ofz  =  2y  +  a  —  xl 

$^  =  3a;2  -  Uxy  +  12y2.      ^  =  -  Gx^ -\-  24xy  -  24y2. 
dx  "^  "^        dy 

dz  ,  dz 

—  =  —  1,     and     —  =  2. 

aa;  dy 

du    dz        _  -       ^,        ,    o^  2       ^^    ^^ 

Hence  u  is  a  function  of  z. 

2.  Is  w  =  l<)g(a;2)  —  2  logy,  a  function  of  2;  =  sinla  +  - j? 

du      2      du  2      dz  /     ,  y\  y       dz  /     ,   v\  1 

— =-,      --=_-,     --=  -co-(a  +  ^).'v     ^=co^(a-f-)— 
dx       x      dy  y      dx  \         xj  x^      dy  \         xf  x 


FUIJCTIONS   OF  TWO   INDEPENDENT  VARIABLES.         121 

du    dz      2         /        y\      du    dz 
dx    dy      x^         \        xj      dy    dx 

Hence  w  is  a  function  of  2;. 

3.  Is  u  ==  x"^  +  y^  a  function  of  0  =  tan  {x  -{-  y)1 

^  =  2^,   J  =  2y,   J  =  sec^x  +  y),    J  =  sec^^  +  y). 

du    dz       ^        „,  .  ^      du    dz       ^         0/      .      \ 

.  • .    -7 —  =  2.r  sec2(a:  +  y),     and     —-•-—  =  2y  sec^far  +  2/)» 

dx    dy  ^  '  dy    dx  ^ 

Hence  u  is  not  a  function  of  z. 


Development  of  Functions  of  Two  Independent  Variables* 

93.  Prop.  To  extend  Maclaurin's  Theorem  to  functions  of  two 
independent  variables. 

If,  in  the  general  development  of  F{x  +  h^  y  +k)^  we  make 
a:  =  0,  and  y  =  0,  and  denote  the  particular  resulting  values  by  the 
use  of  the  [  ],  changing  h  and  k  into  x  and  y  respectively,  we  shall 
obtain 

[d.'^a  "I     xy       VdhfX       y"^        rd'^u~\  x? 

dMy\  '  T  "^  LT^J  *  TT2  "^  Ld^J  '  1.2.3 

Example.  Expand     u  =  e^sin  y. 
du  .  du  dhi  .  d^u 

d^U  .  (^3?/  ,  rf3^ 

^-^^=..-e'smy,    _=-e'cosy,     &c.,     &c. 


122  DlFFkRENTIAL    CALCULUS. 

-M =..£].«,  [|]-..K]-.LS-]='. 

r^i_o  [—1-0  [—-1-1  f-^^l-o 

-7-^  I  =  —  1,  &c.,  the  law  being  quite  apparent. 
x^y         y^  x^y  xy^ 


12      1.2. 3  '    1.2.3       1.2-3 

x^y 


12. 3. 4 


+  &c. 


94.  In  a  similar  manner  we  might  apply  the  general  formula 
deduced  in  the  last  proposition  to  the  expansion  of  any  function 
of  two  variables,  x  and  y,  but  among  these  functions  there  is  one 
of  peculiar  interest,  in  consequence  of  its  frequent  occurrence  in  the 
application  of  the  Calculus  to  Physical  Astronomy.  The  formula 
for  the  expansion  referred  to,  is  known  as  Lagrange's  Theorem. 
It  will  be  deduced  in  the  next  proposition. 

Prop.  Having  given  u  ■=  Fz,  and  z  =  y  -{-  x(pz,  where  F  and 
9  denote  any  function,  and  y  is  independent  of  x,  to  expand  u  in 
terms  of  the  ascending  powers  of  the  variable  x. 

We  observe  first  that  u  \s  a  function  of  x,  and  therefore  if  we 

denote,  as  usual,  by  [w],     7-    ,     -j-^  1  &c.,  the  particular  values  as- 

du  d  Uj 
sumed  by  u^  y^yi?  <^c.,  when  a;  =  0  we  shall  have,  by  Maclaurin's 

Theorem, 

Now  since  «  =  y  +  x^z^ ( 1 ). 

.  • .  when        a;  =  0,     [z]  =  y,     and     .  * .  [w]  =  Fy. 


Lagrange's  theorem.  128 

du       du    dz  du      du    dz 

dx  ~  dz    dx  dy  ~  dz    dy 

But  by  differentiating  (1)  with  respect  to  x  we  get 

dz                    d(pz    dz        ,             dz              cpz  ,^. 

=:^z  +  x-—>—     whence     ^  = ^...(2). 

(13/  UZ         ilJU  UJy  -  U/^)Z 

1  X  •       = 

dz 

And  by  differentiating  (1)  with  respect  to  y  we  havf> 

dz  d(pz    dz  dz  1  ,  . 

dy  ~  dz     dy  ^y  ~  ^  ^f>^   .      .   \»    . 

.Dividing  (2)  by  (3)  and  reducing,  we  obtain 

dz  _        dz 

dx~         dy' 

du       du    dz       du  dz  du 

'    '  dx  ~  dz    dx~  dz  dy  ~        dy 

Hence  when     a:  =  0,     and     z  =  y^  I  — -  I  ==  cpy  — -^. 


-T  .     .  du       du. 

Now  assume  w,  such  that     (p^  •  -7-  =  -—■ 

dy        dy 


Jdu, 
du       du^  dP-u        d'^u^         d^Uy^ 


and 


ft) 


'   '  dx        dy  dx"^       dydx        dxdy  dy 

But      ^  =  ?^.^  =  ^.^^.^=^^.f^  =  (,2)2.^. 
dx         ,dz     dx         dz  dy  dy         ^     '      dy 

'  dx^  dy  Ldx'^J  dy 

A.nd  similarly  it  may  be  shown  that 


UarO""  dy^  '      UxO~  £^^3 


,  <fea 


124  DIFFERENTIAL   CALCULUS. 

But  to  show  that  this  law  of  formation  of  the  differential  coeflS. 
cients  is  general,  suppose  that  it  has  been  proved  that 

d^-H  _  L^     ^         ^^J    .  .  .  .   (4) 


_  .     ,      ^    du       du„_-,  d'^-^u       c?"~%„_i 

Put  {:o'A  »-i .  —  =  — ^-  .  • . — "—* 

^"^^         dy         dy  'dx^-^         Qf.y«-i 

d^U  d'^Un—^  d'^Un-i 


'-m 


'  dx""        dy^'-'^dx        dxdy''-^  dy''-'^ 

ax  dz      dx         dz  dy  dy  ^  dy 


..«  '^'-[(-)-a 


_  ^ ^ (5) 

*  dx""  dy''-^ 

Thus  the  form  (4),  if  true  for  any  value  of  n,  is  also  true  for  the 
next  higher  value.  Now  it  has  been  shown  true  for  n  =  1  and 
/I  =  2 ;  and  hence  it  is  true  when  n  r=:  3,  w  =  4,  &;c.,  or  it  is  uni- 
versally true. 

Now  making  x  =:  d  and  z  ~  y  and  the  expression  (5)  becomes 

Ldx"  J  (/;/"-! 

Making  the  substitutions  for  [?/],    I  -i-    '      y-^  L  <^c.,  in  the  expan- 
sion {A),  we  get 


4(-)^-f] 


+ -^.F^ -lis +*«••••  (^)- 


This  formula  is  called  Lagrange's  Theorem. 


LAGRANGE'S  THEOREM:.  125 

Coi\  Let        u  =  Fz  =  z)     then     Fy  =  y,    and     — -  =  1. 

A  formula  foi  the  expansion  of  z  when  we  have  given  z  =:  y  -\-  xapz. 


EXAMPLES. 

95.  1 .  Given  z^  ^  az  +  b  =  0  to  express  z  in  terms  of  a  and  b. 

Here  z  =  ~  -}-  -  z^,  which  corresponds  to  the  form  z  =  y  -\-  iP(pz, 

when  we  make 

-  =  y,    -  =  a;,     and     2^  =  cpz. 
a  a 

Hence  by  substitution        <py  =  y^  z=  —) 

ay  ay  "^  a^        dy^  dy^  ^  a^ 

Introducing  these  values  into  the  formula  (M),  it  becomes 
b     P  I        b^      1  b''        1  b^         1 

=^[X  +  ^  +  3^  +  12^,+  55^,&c.] 
a"-         a-^  a^  a^  a^^        -' 

If  b  be  very  small  in  comparison  with  a  this  series  will  converge 
very  rapidly. 

2.  Given  q^  —  z-{-z'^  +  z^  +  z^-{-  &c., 

to  revert  the  series,  that  is  to  express  z  in  terms  of  y. 
By  transposition,      z  =  y  —  (z^  -\-  z^  -\-  z*  -{.  &c.) 
Put  a;  =  —  1.     92  =  22  ^  23  ^  g4  _|.  <S2c, 


126  DIFFERENTIAL  CALCULUS. 

Then  (py  =  y^  -\-  y^  +  y*  -{-  6zc. 

d[{(pyy]  ^d[(y^-{-y^-hy*+&,c.Y] 
dy  dy 

=2(2y3+5y*+9ys+ 143/64-  &c^ 
(9y)3  =  (2/2  4.  y3  4.  y4  4.  &c.)3  =  2^«  +  3y7  4-  6y8  4-  <fec 

.  • ,  "^'^^f^^'^  =  5  .  6y*  4-  3  . 6 .  7y5  4-  6 .  7 .  8y6  4-  &c. 

(py)4  =    (y2  4.  y3  4.  y4  4.  &c.)*  =  y^  4.  4^/9  4"  &<2- 

...  "^'[(^-f)']  z=  6 .  7 .  Sy'*  +  4  .  7 .  8  .  9y6  4-  &c. 
dy^ 

((py)5   =    (2/2  4.  y3  4_  y4  4.  &e.)^  =  ylO  +  &C. 

.-.  ^^i^^^  =  7.8.9.10y6  4.&c.        &c..  &c. 
dy^ 

, ' .  By  substitution  in  formula  (i/). 

^  =  y  —  J  [y2  4-  y3  4.  2^4  4_  y5  4_  y6  4_  (S^e.J 

+  ^^^[2.2^3  +  2.5i/^  4-  2.9y«  +  2.Uy^  +  &c.] 
~.Y-i-3[5.6y*  +  3.6.7y5  4-6.7.8y6  4-&c.] 
^         [6  .  7  .  8y5  4.  4  .  7  .  8  .  93/6  4-  &c J 


1.2.3.4 
1 


[7.8.9.IO3/64-&C.] 


1 .2.3.4.5 

-f-  <^c.  =:  2/  —  y"^  -\-  y^  —  y*  +  y^  —  y^  -h  <S2c. 
8.  GiA  en  1  —  s  4-  «*  =  0  to  expand  z". 
Here    z  =  I  -^  e*.     Put  a?  =  1,  y  =  l,(p2  =  e*,  i^g  =  «*, 

dFy  diy*) 

.  (py  =  ev.  Fy  =  y",  9y  •  -^  =  ev  •  -^  =  .i/y-^  cv  =  ne. 


lagranue's  theorem.  127 

d!/  V^^^       dy  ]~  dy 

=  2ne^y .  2/"-^  +  n{n  —  \)  e^v .  2/«-2  =zn(n  +  \) e\ 

dy^L^^^^   '    dy  J~  dy^ 

=  9ne^y  .  y"-!  +  Gn{n  —  1)  e^y  .  2/n-2  4.  ^  („.  _  j)  („  _  2)  c^V   y»-» 
=  ?i  (ri^  -f-  3/1  -}-  5)  e^.     &c.  6zc. 

Hence,  by  substitution  in  formula  (Z),  we  have 

4.  Given  ;?  =  2/  4-  ^ .  sin  ^,  to  expand  ^  and  sin  e. 
Put  a;  =  e,  90  =  sin  z,  Fz  =  sin  2. 

.  • .  (py  =  sin  y ,  (^y)^  =  sin  "^y^  ('^yY  =  sin  ^y  &c. 

.  • .  — —;         =  2  sm  2^ .  cos  y.  =  sm  2y. 

rf2  [((pz/)3"|       d (3  sin  ^y.cosy)        „   .  „         «   .    , 

-^^  =  -^— ^ ?^=6sm2,.cos'2,-3sm>y. 

=  3  sin  2/  (1  4-  cos  2^/  —  -  -f  -  cos  2y) 


=  gsm  y  +  2 ^2  ^'^ ^2/  -2^'^yj 


9  3 

=  -  sin  01^  —  -  sin  y.     &c. 


Hence  by  substitution  in  (M). 
«==:y4-siny54-sin2yj^4-(-sin3y-?sinyj— 1-^+&0. 

A     .        rr  .  dFy        .  1 

Again     Fy  =  sin  y.     .  • .  (py .  -—  =  sm  y .  cos  y  rr:  -  sin  2y. 


128  DIFFERENTIAL   CALCULUS. 

df  dFyl      <? (cos?/,  sin 2?/)       \2       ^  V 


[«■■?] 


dy  L  dt/  J  dy  dy 


c?|-cosy  —  -cos3yj 


3   .    „         1    . 

=z-sm3y--siny. 


dy  4c  4 


^r        3   dFyl       c^Mcos.y.sin3,)  _^^[^"^^^-^"^^-^)] 
dy^  L^^^^   *   dy  J  rfy2  ^^2 

c?2  I-  sin  22/  —  -  sin  4y  j 


_  2  sin  4y  —  sin  2y.     &c.  &c. 


Hence  by  substitution  in  formula  (L). 

1  e       /8  1  \    ^2 

sin  2  =  sin  y  4-  -  sin  2y  .  Y  +  (-sin  Sy  —  -  sin  yj  j-^ 


+  (2  sin  4y  —  sin  2y )  - — ~— -  +  &c. 
Qii    Ji         d  fj     til  w  77  h 

5.  Givenw4-— .-4--T-^'r-^+  :rT  '  i    o   »  +  ^^-  =  0,  to  find 
c/o;    1        dx^    1,2      a^-*     1  .  2 .  3 

A  in  terms  of  w  and  its  differential  coefficients. 
_,  dti  dHi  d^u 

j^i       i?i  \  1 . 2        1.2.3  / 

w  1  ,  -       Vofi^  ,    Pa^^      .    . 

•••2'  =  -^'  =  "^'       '*'=*     T^+T^"^    " 

■  V^%i'   1-2^         V  1.2.3 +  *"7 

Now  if  a  be  a  root  of  the  equation  u  —  0,  and  .r  an  approximate 

value  of  a,  so  that  .t  +  A  =  a,  we  may  use  this  series  in  finding  a 

g 
more  exact  value  of  x.     Thus,  if  re  =  -  =  1  .  5  be  an  approximate 


MAXIMA  AND   MINIMA.  129 

root  of  the  equation     u  —  z'^  —  2x^  -i-  4x  —  8  =  0^ 
then  1  .  5  4-  ^  =  a     and 

23 
Herew=— —        .-.    h  =  .  i\    and   a  =  1 .  5  +  .  11  =  1  .  61 

nearly.     And  if  we  repeat  the  operation  by  putting  x  ~  1 .  61,  a 
nearer  approximatiim  will  be  obtained. 


CHAPTER  X. 

MAXIMA    AND    MINIMA    FUNCTIONS    OF    TWO    INDEPENDENT    VARIABLES. 

96.  A  function  u  of  two  independent  variables  x  and  y,  is  said  to 
be  a  maximum  when  its  value  exceeds  all  those  other  values 
obtained  by  replacing  x  hy  x  ±:  h  and  y  hy  y  dc  k,  when  h  and  k 
may  take  any  values  between  zero  and  certain  small  but  finite 
quantities ;  and  u  is  said  to  be  a  minimum  when  its  value  is  less 
than  all  other  values  determined  by  the  conditions  above  described. 

97.  Prop.  Having  given  u  =  F(x,  ?/),  when  x  and  y  are  inde- 
pendent variables,  to  determine  the  values  of  x  and  y  which  shall 
render  u  a  maximum  or  minimum. 

Suppose  X  to  receive  an  increment  ±  k,  and  y  an  increment  ±  ^, 
the  value  h  and  k  being  small  but  finite  and  entirely  independent  of 
each  other  ;  and  denote  by  Wg  the  value  assumed  by  u,  so  that 

Mg  =  F(x  ±  h,     y  ±.  k). 


130  DIFFERB]NTIAL   CALCULUS. 

Then,  by  Taylor's  Theorem,  as  applied  to  functions  of  two  inde- 
pendent variables,  we  have 

_       ,du    {±h)      dii    {±k)    ,   d^u     (±hY 

^2-^  +  ^*—]"   dy  ~T~'^'d7^"nr 

d^u     (±h)     (±k)       d^u    {±:ky 

'^d^i~\       \~^  dif'~~v:¥"^^^' 

Now  in  order  that  u  may  exceed  Wg  ^^"^  ^  small  values  of 
\  and  A-,  whether  positive  or  negative,  it  is  obviously  necessary 
to  have 

^    (±^    .   ^    (^^)       d'^u    {±hY        dhi     {±h)    {±k) 
dx        \  dy        \  dx^       1  . 2  dxdy        1  1 

in  which  series  we  must  be  at  liberty  to  make  h  and  k  both  positive 
or  both  negative,  or  one  positive  and  the  other  negative  :  or,  finally 
either  may  be  taken  equal  to  zero,  the  other  remaining  finite. 
Now  when  ^  =  0  the  series  (1)  reduces  to 

in  w^hich  h  may  be  taken  so  small  that  the  sign  of  the  first  ternj 

—  •  ,  which  contains  the  lowest  power  of  A,  shall  control  the 

sign  of  the  series.     But  this  term  obviously  changes  sign  with  A, 

since  —  does  not  contain  h ;  and  as  we  are  at  liberty  to  make  h 

alternately  positive  and  negative,  it  is  impossible  that  the  series  (2) 

should  remain  negative  so  long  as  —  •  -^ — -  has  any  value  other 
than  zero. 

We  have  then,  as  a  first  condition  necessary  to  render  u  a 
maximum, 

du    (drzh)       ^  .      .       du       ^  .  ^^ 


MAXIMA  AND  MINIMA.  131 


Omitting  the  first  term  in  (2)  we  have 


dx^      1.2      '  dx^    1.2.3 


+  &c.  <  0  .  .  .  .  (3). 


Here  again  the  sign  of  the  series  will  depend  on  that  of  the  first 
term  when  h  is  small,  and  since  that  term  does  not  change  sign 
when  we  substitute  —  h  for  -(-  ^,  the  series  (3)  will  remain  negative 
for  small  values  of  A,  when 

—  •  \^  <  0,      or  simply  when      ^  <  0. 
Hence  _<  0  ....  (5) 

i 

is  a  second  condition  necessary  for  a  maximum. 

98.  Returning  to  the  series  (1)  and  supposing  A  =  0  while  k  re- 
mains finite,  we  prove,  by  a  course  of  reasoning  entirely  similar, 
that  the  following  conditions  are  also  necessary,  viz. : 

1  =  0....   (C)      and      5<0....(2)). 

Now  omitting  the  terms  in  (1)  which  contain  the  first  powers  ol 
fe  and  k,  and  which  it  has  been  seen  must  reduce  to  zero,  we  obtain 

d-'u    (dbhy        d^u    (zth)    (ztk)      d^u    (±ky      d^u    (=bA)3 
d^'     1.2     ~^d^'      1      '      1  dy^'     1.2    ^  dx^'  l.'Z.Z 

,     d^u      {±h)^.(±k)         d^u_    {±K)(±kY 


'   dxHy  1.2  dxdy"^  1 . 2 

k 
or,  by  making  t  =  ^j  where  r  is  entirely  arbitrary,  since  h  and  k 

have  no  necessary  dependence  upon  each  other. 

l.2Ldx^  dxdy^     dy^J 

h^       fd^u     ^     d^u       ^  „    d^u         ^dH"!       „       ^^  ... 


132  DIFFERENTIAL   CALCULUS 

in  which,  when   h   is   small,  the    sign  of  the   series  will   depend 
on  that  of 


cPu        ^    d^ii         „  d^u 


db2r~—  4-  r 


dx^  dxdy  dy^ 

and  this  must  be  negative  for  all  values  of  r.  whether  positive  or 
negative,  when  w  is  a  maximum. 

^Ve  must  now  search  for  the  condition  necessary  to  rendei 

d^u       .     d'^u  d'^u        ^  /^v   ..       11      1  „ 

-— -  ±  2r- — - — \-  r^-r-r  <  0  .  .  .  .   (5)  for  all  values  of  r. 

dx^  dxdy  dy^  ^ 

d  ii  d  iL  d  it 

Put  for  brevity     -r—-  =  A,      -r— r-  =  B,     and     -r—-  =  C, 
dx^  dydx  dy^ 

Then  A,  B,  and  C,  must,  if  possible,  be  so  related  to   each  other 

that  A  ±  2Br  -j-  Cr"^  shall  be  negative  for  every  roal  value  of  r. 

Now  it  is  known,  from  the  theory  of  equations,  that  if  we  solve 

the  equation  A  ±  2Br  -f  Cr^  =  0  with  respect  to  r,  and  obtain  the 

values 

■=l^B^-JB^  -  AC         ^              z^  B-JB^-  AC 
^\  = -^ '     and     r^z= "L- :, 

and  then  substitute  in  the  polynomial  A  ±  2Br  -f  C'r^,  for  r  values 
alternately  a  little  greater  and  somewhat  less  than  r^  or  rg,  the  sign 
of  fhe  polynomial  will  undergo  a  change.  If  therefore  the  proposed 
substitution  be  possible,  the  condition  A  i  2 Br  -j-  Cr"^  <  0  for  all 
valueB  of  r  will  be  impossible. 

And  so  long  as  the  values  of  r^  and  rg  are  real  and  unequal,  this 
substitution  can  be  made;  but  if  those  values  of  r  prove  imaginary, 
it  will  no  longer  be  possible  to  substitute  for  r,  real  quantities  alter- 
nately greater  and  less  than  such  values,  and  therefore  the  polyno- 
mial cannot  change  its  sign. 

No\^  by  examining  the  values  of  i\  and  rg  it  will  be  seen  that  the 
condition  necessary  to  render  r^  and  rg  imaginary  is  B'^  <C  AC, 
Hence  we  have  a  fifth  condition  necessary  for  a  maximum,  viz.  • 


MAXIMA   AND   MINIMA.  183 

when  this  condition  is  satisfied,  the  condition  (5)  will  also  be  satis- 
fied, since  (5)  is  true  when  r  =  0. 

It  ought  to  be  remarked,  however,  that  when  B^  =  AC,  the  two 
roots  Vi  and  rg  become  real  and  equal,  and  therefore  in  passing 
over  one  of  these  roots,  we  necessarily  pass  over  both.  Thus  the 
sign  of  polynomial  will  not  change,  so  that  the  fifth  condition  would 
be  more  correctly  stated  as  follows : 

dx^'^dy^        \dxdyj  >      •  •  •  •    ^     ^ 
By  a  course  of  reasoning  entirely  similar,  we  can  prove  that  the 
five  conditions  necessary  to  render  u  a  minimum  are  the  following; 

du  du  d'^u  d'^u  d^u    d^tt        /  dhc  \^i=  , 

^  "^ '  ^  "^ '  ^  ^  '  df^'  d^'df~  [dm)  > 

-_     .,,dhi       ^       ^        du       ^      .  -  .  , 

99.  11  -7-r=  0,  when  — -  =  0,  there  can  be  neither  maxmium  nor 

dx^  dx 

Urif  (17/ 

minimum,  unless    .-.  =  0  also;    and  similarly,  if  — -  =  0,     when 

dx^  ''         dy^ 

— -  =  0,  there  can  be  neither  maximum  nor  minimum  unless  - — =  0. 
dy  dy^ 

There  are  other  conditions  likewise  necessary   to   render    u    » 

maximum  or  minimum  in  such  cases,  but  they  are  usually  of  so 

complicated  a  character  as  to  be  unfit  for  use. 

EXAMPLES. 

100.  1.  To  determine  the  values  of  x  and  y  which  render 
u  =.  x^  ■\-  y"^  —  Zaxy  a  maximum  or  minimum. 

^  =  3:^2  -  3ay  =  0,  .  .  .  .  (1 ) .       ^  =  3y»  -  3«:«  =  0,  .  .  .  .  (2), 

From     (1),    y  =  — ,  and  this  substituted  in  (2),  gives 

a;*  —  a^a;  =  0 ;       .  • .  a;  =  0,      or,      x  —a, 

the  two  other  roots  being  imaginary. 


/^ 


134  Dlf^FEllENTIAL    CALCULUS. 


it 
But  when  a;  —  0,     y  =  —  =  0, 

and       when  x  =.  a^     2/  =  «• 

Now  forming  the  second  differential  coefficients,  we  get 

dP'u  d^u 

- —  =  6a;  =  0     when  a:  =  0,        -j-r  =  Qy  =  0     when  V  =  0, 

=.  6a  when  x  =  a,  =  Qa  when  y  =z  a, 

d^u  ^      d^u    dH       /  dhi  \2 


d^dy  =-^^Txi-df~  l^)  =  ~'"'  "'''"  .  =  0  and  2,  =  0. 

=  27a2  when  a;  =  a  and  y  —  a. 
.  • .  The  five  conditions  necessary  for  a  minimum  are  fulfilled 
when     X  z=i  a     and     y  =.  a^  viz. 

.  • .  u  =:  a^  -\-  a^  —  3a^  =  —  a^. 

But  when  a:  =  0  and  y  ==  0,  -— r  and   -7—  reduce  to  zero,  while 

dx^  dy^ 

UrlA  d   U 

~rr-^  and  -r-^  do  not  reduce  to  zero.     Hence  the  value  w  =  0,  is 
dx^  ay^ 

neither  a  maximum  nor  a  minimum. 

2.  To  find  the  lengths  of  the  three  edges  of  a  rectangular  par- 
allelopipedon  which  shall  contain  a  given  volume,  a',  under  the 
least  surface. 

Let     a;,  ?/,  and  z,  be  the  required  edges,     .  * .  xyz  =  a^  .  .  .  (1). 

And     u  =.  2(xy  -{-  xz  -\-  yz)  =  surface  =  a  nainimum. 

a^  a? 

But  from   (1),     xz  =  — ,  and  yz  z=i — » 


i^y  +  j+^)----m- 


du       ^          2a?        ^          ,_.         .du       _          2a3  . 

5J  =  2y--^  =  0,...  (3).  and  ^  =  2^-^  =  0 (4) 

• .  x^'s  =  a^  =  xy^,     .  • .   X  =z  y^     and  consequently  x^  =  a', 


MAXIMA   AND   MINIMA.  135 

.  • .  a:  =  a,     y  =z  a^     and     z  =z  a. 


dx^     dy"^       \dxdy)  ~  ""'  ^  "' 


12  >0, 


,  • .    -M  =  2(a2  4-  a2  _j_  0^2^  _  ^0^2  _  a  minimum,  and  the  parallelo- 
pipedon  must  be  a  cube. 

3.  Given  x  -\-  y  ■\-  z  i=.  <:( ^  to  find  the  values  of  rr,  y,  and  z^  when 
cos  arcos  y  cos  2  --  z^  =:  a  maximum. 

Regarding  x  and  y  as  independent  variables,  and  z    a  function 
of  X  and  y,  we  obtain  by  differentiating 

X  ■\-  y  ■\-  z  ^=.  t< ^  with  respect  to  x  and  y  successively. 

>+!=«■  ^"^  '+!=« (!)• 

But  since  it  ■=.  maximum, 

log  u  =  log  cos  X  -\-  log  cos  y  +  log  cos  z  =  maximum, 

((/  locr  u\  dz 

— r- — )  =  —  tan  X  —  tan  2  -7-  =  0, 
dx    I  dx 

(d]o2,u\  dz      ^  . 

and  I  — r^—  )  =r  —  tan  y  —  tan  2  --  =  0. 

\    dy     J  ^  dy 

or,  by  replacing  --  and   —  by  their  values  derived  from  equa> 

tions    (1). 

—  tan  x  +  tan  2  =  0,     —  tan  y  +  tan  g  =  0, 

1 

• .   tan  X  =  tan  z  =  tan  y,     and     x  =  y=z=-'Jir. 

o 


3I      /'V    1 


4.  To  find  the  greatest  rectangular  parallelopipedon  which  can  be 
inscribed  in  a  given  ellipsoid. 

Let  a,  h,  and  c,  be  the  semi-axes  of  the  ellipsoid,  x,  y,  and  z^ 
tbp  co-ordinates  of  one  of  the  vertices  of  the  parallelopipedon. 


136  DIFFERENTIAL  CALCULUS. 

Then  2x,  2y,  and  2z,  are  the  three  edges  of  the  parallelopipedon, 
alid,  therefore,  2x  .  2y  .  20  =  maximum,  or 

u  =  xyz  =  maximum (1). 

But,  since  each  vertex  is  in  the  surface  of  the  ellipsoid,  the  co- 
ordinates X,  y,  z,  must  satisfy  the  equation  of  the  surface. 

.••4+1+^  =  1 (2)- 

a^        b^        c^  ^ 

Differentiating  (2)  with  respect  to  x  and  y  successively,  regarding 
g  as  a  function  of  the  independent  variables  x  and  y,  we  get 

2x      2z    dz  _         2y      2z    dz  _ 


But,  from  (1)   we  have 

dz       ^        ln.v\  .         dz 

dy 


0=^^+^-!=*^'  0=^-+-j'-5^.=o. 


dz  dz 

or,  by  introducing  the  values  of  ~  and  —  from  equations  (3). 

^^"^2/^  =  0,      and      xz  -xy'j^  =  0. 


a^z  l)^z 

• .  a^z"^  —  c^x"^     and     h^z"^  =  c^y^     .  • .  — -  =  -^  =  7- 

■     a^       c^       6-* 


x"^       z^       y^ 


it/  if/  i*/  I* 

Hence  from  (2),  — n  H n  H r  =  1      and      x  =  -—=  :    in    like 

^  '    a^       a^       a^  y^ 


h  c 

manner  it  may  be  shown  that,  y  =  — :,  and  z  =  — 

y  3  y  3 


Thus  the  edges  of  the  parallel opipedon  must  be  proportional  to 
the  axes  to  which  they  are  parallel.  In  each  of  the  last  two 
examples,  the  formation  of  the  second  differential  coefficient  has 
been  omitted  as  unnecessary,  it  being  easily  seen  that  the  proposed 
question  admitted  of  the  maximum  or  minimum  sought,  and  also 
that  the  values  found  were  the  only  suitable  values. 


CHAPTER  XL 


CHANGE    OF    THE    INDEPENDENT    VARIABLE. 

101.  Hitherto   we   have   employed    the   differential   coefficienti 

dy    d'^y    „  du  dhc    .  ,     .     ,  ,      ,  i     .      , 

-~,  -— ,  &c.  or  — ,  -j-^,  &c.  exclusively  upon  the  hypothesis  that  x 

was  the  independent  variable.  Dut  there  are  many  cases  in  which  it 
is  more  convenient  to  adopt  some  other  quantity  t  upon  which  both 
X  and  y,  or  x  and  u  depend  as  the  independent  variable,  and  perhaps 
to  pass  from  one  supposition  to  the  other  within  the  limits  of  the 

same  inv^.stigation. 

c??/     d'^zi  du   dP"Uj 

It  then  becomes  necessary  to  express  — ,  — -|,  &c.  or  — -,  — — ,&c, 

dX      QiX  d'Xi     OiX 

in  terms  of  the  differential  coefficients  of  a:  and  y,  or  those  of  a;  and  u 

taken  with  respect  to  the  new  variable  t. 

dv  dP"tj 

102.  P/'ojc.  Given  y  =z  (pa?,  and  x  =  Ft^  to  express  -^,  and  -r-^  in 

,.  dx   yi'^x   dy   d'^y    . 

Since  y  is  a  ivinction  of  ar,  and  x  a  function  of  /,  we  have 

dy 
dy      dy    dx  ,,.  .  dy        dt,  ,  ^. 

di 

dij 

Now  differentiating  (1),  and  observing  that  ~   is  a  function  of  i 


through  X,  we  get 


d^y      d^y    dx^      dy    d^x 
'd^~d^"dfi~^dx"di^' 


(^) 


138  DIFFERENTIAL   CALCULUS. 

dhj      d]i    d'^x      d^y    dx       d'^x   dy 
d'^u  __~di^^'dx"dfi  _'d^'~dt~~dfi"di 
'  dx^  ~  dx^         ~  dx^ 

dfi  dF 

The  two  formulae  {A)  and  {B)  resolve  the  problem. 

d'^u  d^n 
Cor.  In  a  similar  manner  we  might  form  expressions  for  — -,  — ^ 

dx^  dx* 
upon  the  same  hypothesis,  but  they  are  seldom  required. 

Cor.  If  y  be  takeji  as  the  independent  variable,  then 


and      .-.  :i^=l^=0. 


tz 

dy 
=^^di 

dy 
-dy 

=  1 

dy 
'  dx~ 

1 

dx~- 

1 

dx 

dt 

dy 

d^y 
dt'' 

~    dt 
dH 

d'^y 

^2 

=  - 

dy^ 
dx^ 
dy^ 

and 


Cor.  If  x  be  the  independent  variable,  then 

dx       dx  d'^x 

t  —  x^   —-  =  --=  \     and     -—=zO,  and  (A)  and  (B)  reduce 

dy       dy         _  dh/       d^y    . 
to  T^  7~^  ^       IHi  ~  T^'         ordmary  forms. 


EXAMPLES. 

103.   1.  Transform  the  differential  equation 

c^y  ^         dy  y  ^  ,  , 

—r^  — ^  •  -r-  +  -. 5  =  0,  SO  as  to  render  fl  the 

dx'       i  —  x^   dx        I  —x^ 

independent  variable,  having  given  &  =  cos  — ^a?. 

dx  d'x 

Here    x  =  cos  6.     .•.—=—  sin  ^,  — -  =  —  cos  5  =  —  a?, 
do  do^ 

dy 
,     dy  ^  dd  _  1      dy 

'  dx~  dx~       sin  &    dd 


CHANGE  OF   THE   INDEPENDENT   VARIABLE.  139 


d^y 

d'^y    dx       d'^x  dy 

c/a2    dd        dd^  d&          1       d'^y       cos  d    dy 

dx-'~ 

dx^                  sin  2a     dd^       sin^a   dd 

dd^ 

Hence  by 

substitution  in  the  given  equation, 

1 

sin  24 

d^y        cos  6     dy   ^    cos  6    dy  ^       y         ^ 
d6^        sin  3a     d&   '    sin  3a    dd    '    sin^a 

f.+'=»- 

or 

Tliis  example  illustrates  the  important  fact,  that  a  change  of  the 
independent  variable  will  sometimes  simplify  the  form  of  the  differ 
ential  equation. 

2.  Transform  -7— •  -f  -r-^  =  0,  so  as  to  render  r  the  independent 


dx^        dy"^ 

dr\x) 


variable, 

where  1 

r2  = 

x'  +  : 

y^' 

Here 

ar2  = 

r2- 

y'    . 

dx        1 
"*  ~dr^~. 

1 

X 

r 

dx         1 
dr         X 

And  similarly 

dy 
dr 

r 

~  y 

d'^y 
'        C//-2  - 

d^u 
dx^~ 

dx 
dr 

d^x    du 
di^    dr 

*    • 

dx^ 

r       ^     d'^x 


^2      (^j.2  y3      (jIj. 


dr 


.  d'^u       7/2    (Pk,       pc;^    du 

dy^        r2    c/r2        r"^     dr 

dii         oil 
.  • .    By  substitution   in  the  given  relation  -—  +  — -^  =  0,  and 

J      .  d^u       1    du        . 

reduction,  — — ■  H —  •  -~—  =  0. 

dr^       r    dr 

104.   Prop.  Having  given  u  =  F  {x,  y)  when  x  =  cp  {r,  a)   and 

/.  /     ,,^  du        .    du  . 

y  =f  (r,  a),  to  express  —  and   -r-  m  terms  of  r  and  a. 
u2>  uy 


140  \  DIFFEBENTIAL    CALCULUS. 

Since  u  is  a  function  of  x  and  y,  each  of  which  is  a  function  of  r, 

we  have 

du        du    dx       du    dy 

dr        dx    dr       dy    dr 
And  .similarly,  x  and  y  being  functions  of  ^, 
du        du    dx       du    dy 

~a^'^7i"dd'^  Ty"dd ^  ^ 

Multiply  (1)  by  — -,  and  (*2)  by  —  and  subtract;  then  multiply  (1) 

by  -h-  and  (2)  by  -i-  and  subtract.     We  shall  then  obtain 

du    dx        du    dx  du  /dx     dy       dy    dx\ 

dr  '  dd        dd     dr  ~        dy  \dr  '  dd        dr    dd  i 


and 


di(,    dy       du    dy        du  /dx     dy       dy     dx\ 


du    dy       du    dy  du    dx       du    dx 

du       dr    d.)        dd     dr  ^      du  dr    dd        dd     dr 

and      -r-  = 


dx       dx    dy       dy     dx  dy  dx    dy       dy    dx 

dr  '  dJ  ~  dr  '  dd  dr     dd        dr     dd 

105.    These    formulae   become   much   simplified    when    we   have« 
X  z=  r  cos  ^,  y  —  r  sin  ^,  the  common  formula  for  passing  from  rec- 
tangular to  polar  co-ordinates.     For  we  then  have 

dx  .      dy        .    .      dx  -    ^      dy  . 

-—  =  cos  ^,    -r-  =  sm  6,     -—  =  -^  r  .smO,     -^  =:  r  cos  d, 
dr  dr  dd  dd 

dx    dy       dy    dx  .      „.    ,     .  om 

.  • .  —  .  -4- ■;-'-rT  =  r(cos24  -f-  sni^^)  =  r. 

dr     dd        dr    dd  ^  ^ 

du  du       sin  d    du.  du        .       du       cos  d    du 

dx  dr  r       dd  dy  dr  r       dd 

Ex.  liavmg  given  x- y  —  =z  a,  to  transform  the  equation  to 

the  variables  r  and  d,  where  x  =  r  cos  d,  y  =  r  sin  d. 

du  du 

X-- y— -  =  rcos 

dy       ^  dx 


./  .    ^du.       cos^    du\  .      /         du      B\n  6  du\ 

)(sin43-  H —)— rsm^(cosd-3 -jr-l 

\        dr  r      dd)  \        dr         r    d6f 


du 


=  (cos^fl  +  sin«J)-  =  ^  =  «, 


LIBRARY^ 

u:ni  V  Kirs  iTV  op 
CALIKOILMA. 


CHAPTER    XII. 


106.  It  has  been  shown  that  the  general  development  of  F{x  -f-  A), 
so  long  as  the  value  of  h  remahis  unassigned,  is  of  the  form 

F{x  -\-h)  z=Fx-\-  Ah-\-  Bh?  +  (7A3  +  &c (1), 

containmg  none  but  the  positive  integral  powers  of  A. 

But  although  this  be  true  for  the  general  value  of  ic,  it  is  possible 
in  some  cases,  to  assign  certain  particular  values  to  x^  which  shall 
cause  fractional  powers  of  h  to  appear  in  the  development ;  and  to 
such  cases  Taylor's  Theorem  does  not  apply,  because  its  proof  de- 
pends upon  the  assumption  that  the  series  (1)  holds  true.  If,  for 
example,  we  assign  to  x  such  a  value  as  shall  cause  fractional  powers 
of  h  to  appear  in  the  undeveloped  function,  we  may  expect  to  find 
similar  powers  in  the  development,  and  we  therefore  cannot  expegt 
Taylor's  Theorem  to  give  the  correct  expansion.  Now  when  the 
particular  value    a:  =  a  introduced  into  the  undeveloped  function 

the  fractional  power  A",  there  must  have  been  in  the  general  ex- 
pression for  Fx  (before  a  was  substituted  for  x)  a  term  of  the  form 

{x  —  «)»  which  becomes  {x  —  a.  -\- K)"^  in  F{x  +  A),  and  reduces  to 

A*  when  X  z=.  a. 

When  this  occurs  some  of  the  differential  coefficients  will  cer- 
tainh"  become  infinite,  if  we  make  a;  =  a. 


142  DIFFERENTIAL   CALCULUS. 

To  illustrate  this  fact,  take  the  example 

u  =  Fx  =zb  ■\-  {x  —  ay  -\-  {x  —  ay, 
and  suppose  x  to  receive  the  increment  h,  converting  u  into 

m 

u^  =  F{x-{-  h)  =b~{-  {x  —  a-^  hy  +  {x  —  a-{-  hy. 

m  ■ 

Now,  for  the  particular  value  x  =  a^u-^  becomes  b  -{•  h^  ■{■  h\ 
But  by  for.ning  the  successive  differential  coefficients  of  u  with  re- 
spect to  x^  we  get 

dii       „,  „       m  ,  .^-1 

^=  1.2,3 +  !!i(!^-i)(^-2k-«)"-. 

ax-^  n\n  /\n  /  ^ 

It        7W//M        ,\/'/^        ^\i>n        .A,  ^^_4     „ 


and  since  the  exponent  of  x  —  a  is  diminished  by  unity  at  each  dif- 
ferentiation, it  must  eventually  become  negative,  rendering  the  co* 
efficient  infinite  when  x  ■=.  a.  Moreover,  all  the  succeeding  differen- 
tial coefficients  will  likewise  become  infinite. 

It  may  be  observed  also  that  if  the  lowest  (and  therefore  the  first) 
fractional  exponent  which  appears  in  the  deveh'pment,  be  interme- 
diate in  value  between  the  integers  r  and  r  +  1 ;  then  the  first  di^ 
ferential  coefficient  which  becomes  infinite  will  be  the  (r  -f-  \)th. 

It  appears  then  that  this  peculiarity  will  arise  whenever  the  value 

assigned  to  a;  causes  a  surd  (such  as  (x  —  a)"  )  to  disappear  in  Fc^^ 

while  the  corresponding  surd  [(.r  —  a  4- ^' )"  ]  continues  to  appear 
in  F{x  4-  h)  in  the  form  of  a  fractional  power  of  h.  This  inappli- 
cability of  Taylor's  Theorem,  improperly  called  a  failure  of  the 


143 

theorem,  occurs  precisely  when  the  development  is  impossible  in  the 
general  form,  and  therefore  does  not  result  from  any  defect  in  the 
theorem  itself. 

Again,  it  has  been^shown  that  the  general  development  does  noi 
contain  negative  powers  of  /i,  because  we  would  have,  (if  there  were 
such  a  term  Ch~'^)  F{x  4*  ^0  =  ^^  —  ^  when  A  =  0,  an  obvious 
absurdity.  But  when  we  assign  to  x  such  a  value  a  as  shall  render 
Fx  T=z  CD ,  the  above  argument  ceases  to  be  conclusive.  In  this  case 
Fx  =  CD  ,  and  the  differential  coefficients  will  be  infinite  also.  Thus 
Taylor's  Theorem  will  be  inapplicable. 

Here  also  we  see  that  the  presence  of  a^ negative  power  of  h  in 

the  development  must  result  from  a  term  of  the  form —  in  Fx, 

^  (x  —  a)«  * 

which  becomes -— r-  in  Fix  -}-  h)  and  reduces  to  -r-  =  Bh-^ 

{x  —  a  -\-  h)  ^  '  A" 

when  a;  =  a. 

We  conclude,  therefore,  that  there  are  two  cases  in  which 
Taylor's  Theorem  is  not  applicable,  viz. : 

1st.  Wiien  x  =ia  causes  a  surd  to  disappear  in  Fx^  thereby  in- 
troducing a  fractional  power  of  h  into  F  (x  -{-K), 

2d.   When  x  =.  a  renders  Fx  =  ao . 


EXAMPLES.. 

107.  1st.  Case.  Given    u  =  b  -^  {x -^  c)^ -\-  {x  -  ay=  Fx,      tc 
expand     Uj^=:  F(x  +  h)  =  b  +  (x-^  c  i-  A)2+  (^x  —  a  +  hy. 


^__1    1    3  -i 

.  • .  ]\y  substitution  in  Taylor's  Theorem, 


144  DIFFERENTIAL   CALCULUS. 

M,  -_.  5   f  (^  +  cY  4-  (^  -  a)^+  \2{x  +  c)  +  l{x  -  af\j 

Now  this  development  is  entirely  true  for  all  values  of  x,  except 

X  =  a^  v/hich  renders  the  term     1  •  2  +  -  •  -(a;  —  a)        — — ,  and  all 

suoceed'./]g  terms,  infinite  ;    the  true  development  in  this  ease  being 

«i  =  6  +(a  +  c  +  /O'  +  ^^*-'=  6  +  (a  +  c)2  +  2(«  -f-  c)h  +  h^  +  h^, 
which  agrees  with  the  series  (1),  only  so  far  as  to  include  the  term 

h 


[2(:.4c)+|(.-«f]i 


2d.  Case.  Given     u  =  b  -\   vun  a:  -^  - — - — -  =  Fx,  to  expand 

[x  —  ay 


c 


u, ^ F{x -i-h)=b  +  M-  +  '^^  H-  (^z:^_pip- 

rftt  1  .  2c         rf%  1  .  2  .  3c 

— -    =  cos  a;  — rr?       -J-.-  -.■   -  •  Sin  ;r  -f  ;r , 

dx  (x  —  a)3       ^A"'  {\  - '  a/ 

c/3«^  1.2.3.4c  .  . 

-— -  =    —  cos  X ;; :t— »  WC  ,  (WC 

dx^  (x  —  a)^ 

.'.  By  substitution  in  Taylor's  Theorem, 

c      r         1.2c  lA 

„  _  6  +  sin  X  +  ^_r^,+  [cos^  -  -^--^,}j 

r       .        .1.2.3C-1    7*2        r  l.l\3.4c-l    ^3 

4-       —  Sm  a:  4- r— r  +      —  COL'  r.  -  •  -7 —     ——5  &c 

L  (^  —  «)  J  1-2       L  (a;  —  a/  J  1 .2.3 

This  development  is  correct  except  when  x  =z  a,  the  true  devel 
opment  then  being  (Art.  48) 

c  k'^ 

u  ~=  b  -\-  sin  (a  -\-  h)-^  -—  —  6  +  sina  +  cA"-2-|-cosa./i— sina— — .  (fee 
h^  1.2 


FAILURE    OF   TAYLOR'S   THEOREM.  145 

flere   the   very   first    term   given   by  Taylor's   formula,  viz.: 

Fx  =:b  -^  sin  a  -f-  7 :^'     is  incorrect. 

(a  -  af 

108.  Prop,  if  the  true  development  of  F(x  -f  h)  contain  posi. 
tive  integral  powers  of  h  to  the  (n  —  \)th  power  inclusive,  followed 
by  a  term  containing  h*  where  s  is  a  fraction  intermediate  in  value 
between,  n  —  \  and  w,  the  first  n  terms  of  the  expansion  will  be 
given  correctly  by  Taylor's  Theorem,  but  the  {n  +  \)th  term  will 
not  be  given  correctly. 

Proof.  Let  the  true  development  of  F{x  -\-  h),  when  x  =  a,he 

F(x  -{-  h)=  A  +  Bk  +  Ch^  -\-  Dh^ +  Nh^-^  -f  Ph»  +  &c., 

where  s  denotes  a  fraction  greater  than  n  —  \  and  less  than  n. 

Then,  since  the  diflTerential  coefficients  of  F{x  -{-  A),  taken  first 
with  respect  to  2:,  and  afterwards  with  respect  to  A,  are  equal,  we 
have 

ax  ah 

■\-{n  —  l)iV^A'»-2-f  sPh'-^-^  &c. 

1.2C4-2.3Z>A 

+  {n-  Z){n  - 

+  (5  —  2)(5  —  \)sPh'-^  -f-  &c. 


ax^  dh^ 


"IlE^JlI  =  l,2.ZD +  {n-  3)(w  -  2){n  -  l)Nh»-i' 


-\-(s—n+2){s-n+S) (s— 2)(s— l)sPA»-«+i-i-<Sca 

d^F{x  +  h) 


dx" 


=  {s  -n  -i-  l)(s  -n-\-  2)(s  —  w  +  3) . 


{s  -  2){s  -  l>PA»-«-f  &c. 
Now  when  h  =  0,  the  preceding  expressions  reduce  to 

10 


146  DIFFERENTIAL  CALCULUS. 

-^^  =  1.2.3 («-3)(«-2)(»-l)iV. 

-^  =(.-n+l)(.,-«+2)(«-«+3) (.-2)(s-l>  -  =  « . 

„      dFx       „         1    d^Fx      , 

Thus  each  of  the  terms  A,  Bh,  Ch^,  &c.,  of  the  true  development 
will  be  given  correctly  by  Taylor's  Theorem  as  far  as  the  term 
Nh'^~^  inclusive  (that  is  to  n  terms),  but  the  {n  ■\-  V)th  term  of  the 
true  expansion  is  Ph'^  while  by  Taylor's  series  it  would  appear  to 
be  infinite. 

The  results  established  in  th;"s  proposition  are  important,  because 
it  frequently  occurs  that  the  first  or  leading  terms  of  an  expansion, 
are  those  only  which  we  have  occasion  to  consider. 


PART  11. 

APPLICATION  OF  THE  DIFFERENTIAL  CALCULUS  TO 
THE  THEORY  OF  PLANE  CURVES. 


CHAPTER    I. 

TANGENTS    TO    PLANE    CURVES. NORMALS. ASYMPTOTES. 

109.  In  the  application  of  the  Differential  Calculus  to  the  investi- 
gation of  the  properties  of  plane  curves,  we  regard  the  two  variable 
co-ordinates  x  and  y  or  ^  and  r,  which  serve  to  fix  the  position  of  a 
point  on  the  curve,  as  the  independent  variable  and  the  dependent 
function  respectively. 

These  two  quantities  are  connected  by  a  general  relation  called 
the  equation  of  the  curve. 

Such  as     y  z=z  Fx    OT    r  =  cp&,    F{x,  y)  =  0,  .  or    (p(r,  d)  =  0.  . 

When  the  form  of  this  equation  is  given,  we  can  readily  deter- 

dy    dp"  XI 
mine   the  values   of  the   differential   coefficients    — ?  -7-^'  &c.,  or 

(aiX     clx 

df    dP'v 

Ta'  -TTT5  <^c-)  hi  terms  of  the  co-ordinates,  and  these  values  will  be 

found  extremely  serviceable  in  the  discussion  of  the  properties  of 
ihe.  curves. 

110.  The  first  application  of  the  Calculus  to  Geometry  which  it 
is  proposed  to  make,  is  the  determination  of  the  tangents  to  plane 
curves. 


148 


DIFFERENTIAL   CALCULUS. 


Prop.  To  find  the  general  differential  equation  of  a  Hue  which  is 
tangent  to  a  plane  curve  at  a  given  point  x-^^  y,. 


O  TAD.  Da  X 

The  equation  of  the  secant  line  RS,  passing  through  the  points 
r^  2/1  and  x^  y^-,  is 


y  -yi  = 


\x-x,).,  ..(1). 


But  if  the  secant  RS  be  caused  to  revolve  about  the  point  Pj,  ap- 
proaching to  coincidence  with  the  tangent  TT,  the-  point  Pg  ^'^^ 
approach  Pj,  and  the  differences  y^  —  y^  and  X2  —  arj  will  also  di- 
minish, so  that  at  the  limit,  when  RS  and   TV  coincide,  — — 

X2  •—  x^ 

will  reduce  to  -^,  and  the  equation  (1)  will  take  the  form 

dy. 


y  -y\ 


dx. 


(x  -  x^) 


r  (2), 


which  is  the  required  equation  of  the  tangent  line  at  the  point  x^  y,. 

111.  To  apply  (2)  to  any  particular  curve  we  substitute  for 
-—-  its  value  deduced  from  the  equation  of  the  curve  and  expressed 
in  terms  of  the  co-ordinates  of  the  point  of  tangency. 

Oor.  The  differential  coefficient  —^  represents  the  trigonometrical 

ClX-i 

tangent  of  the  angle  P^TX  formed  by  the  tangent  line  with  the 
axis  of  X. 

Cor.  To  find  the  value  of  the  subtangent  D-^T^  we  make  y  =  0 
in  (2).     The  corresponding  value  of  x  will  be  the  distance  OT,  and 


TANGENTS  TO  PLANE  CURVES. 


149 


therefore  x  —  x^  will  represent  the  subtangent  i>i^,  this  latter  being 
reckoned  from  D^  the  foot  of  the  ordinate.     Thus 


subtan  D-^T  =z  x 


X.   — 


dx-. 


(3). 


In  the  formula  (3),  x  represents  the  independent  variable,  but  if 
we  take  y  as  the  independent  variable,  this  formula  may  be  simpli- 

fied.     For  it  has  been  shown  that  ~  =  — -—    or    -p-  =  -;-.     Henco 

ax        ax 


(3)  may  be  written 


dy 


ily^      dy 
dx 


dx. 


subtan  i)ir=  -yi~ (4). 

112.  Prop.  To  determine  the  general  differential  equation  of  a 
line  which  is  normal  to  a  plane  curve  at  a  given  point  i^i  yj. 

The  equation  of  the  normal 
FN,  which  passes  through  the 
point  a?!  ^1,  will  be  of  the  form 

y  -yi  =  h(x-x^)  .  .  .  (5), 


O  T      A     D 


where  t^  denotes  the  unknown 

tangent  of  the  angle  PNX  formed  by  PN  with  the  axis  of  x. 

But  since  the  normal  PN  is  perpendicular  to  the  tangent  PT^  we 
must  have,  by  the  condition  of  perpendicularity  of  lines  in  a  plane, 


1  +  //,  =  0     or     ^  = where     t  =  -p- 

*  ^  t  dx^ 

Replacing  t^  by  its  value  in  (5)  there  results 


tan.  ancle  PTI\ 


dxi 


-^)=-S-(^--')  ••••(«)• 


dx^ 


To  apply  (6)  we  substitute  for  — -  its  value   derived  from   th« 
equation  of  the  given  curve. 


/ 


150 


DIFFERENTIAL   CALCULUS. 


.   Cor.  To  find  the  value  of  the  subnormal  i>iV,  we  make  y  =  0 
in  (6)  and  thus  obtain  ON  as  the  corresponding  value  of  x. 

.•.DN=x-x,  =  y,^^^ (7), 

when  either  the  subtangent  or  subnormal  has  been  determined,  the 
tangent  and  normal  can  be  readily  constructed. 


APPLICATIONS. 


113.  1.  Let  the  curve  be  the  common  parabola,  whose  equation  is 
y^  =  2px. 

...^=:^,     ^=Z.,     and     ^-^. 
'    '  dx      y      dx-^       y^ 


dy^ 


Hence  the  equation  of  the 
tangent  is 

{x  -  x^) 


y  -Vx 


yi 


or  yyi  -  yi"  =  p(x  -  x{), 

whence  t  o        d        n 

And  that  of  the  normal  is 

y-yi  =  -j(^-  ^i)- 
y\  ^p^i 

P  P 


Also, 


subtani^J": 


and 


P 


subnorm  DN  z=  y,  —  z=  p. 


Thus  it  appears  that  the  subtangent  of  the  parabola  is  negative  and 
equal  to  twice  the  abscissa ;  and  the  subnormal  is  positive  and  con 
stant,  being  equal  to  the  semi-parameter.' 


TANGENTS  TO   PLANE   CURVES. 
2.  The  Ellipse,  aV  4.  hH"^  =  anj"^ 


151 


dx 


. . .  ^1  ^  _       1       and     -j^ 
dx^  a?y^  dy^ 


.  • .  The  equation  of  the  tangent  is 


'i;» 


or,         CL^yVi  +  b^xx-^  =  a^-b'^. 


dxi      o^y-i" 


Also  subtangent  =  —  y^  — -  = 


dy^        hH^ 


= X., 


And  subnormal 


yi 


dx^ 


b^ 


o.  The  logarithmic  curve,  whose 
equation  is     y  =z  a'. 


dx 


=  log  a  .  a*. 


subtan  =  — 


=  — wi. 


log  a  .  a*i  log  a 

where   m   is    the   modulus   of    the   system   of   logarithms   whose 
base  is  a. 

Also    subnorm.^::  log  a .  a'^^y^ 


In  this  curve,  the  values  of  the  abscissas  are  the  logarithms  of  the 
values  of  the  corresponding  ordinates  in  the  system  whose  base  is  «. 

114.  Pro'p.  To  determine  expressions  for  the  tangent,  the  normal, 
and  the  perpendicular  from  the 
origin  to  the  tangent  of  a  plane 
curve. 

^'or  the  tangcLt  P  T,  we  have 


PT  =  '^PD''  -i-  £>T^ 


14 


^ 

^^1^* 


162  DIFFERENTIAL   CALCULLS. 

For  the  normal  PIS',  we  have 


PN=:  -y/FD^  +  DN^  =  y.yji  4-  ^- 
For  the  perpendicular   0^,  we  have 

Ex.  The  general  equation  of  all  parabolas. 

The  general  name  of  parabola  is  applied  to  all  curve?  included  in 
the  equation  y'^  =  a'^—'^x,  in  whi'.  h  m  may  7'epresent  any  positive 
number  either  whole  or  fractional.  When  m  =  2,  thf*  curve  be- 
comes the  common  parabola. 

TT  -       «  1  dy     ■   ^^~^  3     d^       m?/'^'^       mx. 

Here    y^=a^-^x,      .'.  —  =  -?     and     -—  =  — ^ —  =  — L 

^  dx       my"^-^  dy         a'»  ^  y^ 

.  dx.  m?/,*" 

dy^  a""-!  ' 

dy^         a'"-^  .y^2 

subnorm  =y.~  =: -—  =  — i-, 

^^dx^       my{^-^       mx^ 

tan  =  y,  a/  1  +  ^2  =  ^/y^+  niW, 
»d  perp  =  ..^^-^^.  =  iMLllJ!^^ 


D+5-T  [^.^+»wj* 


115.  Prop.  To  obtain  expressions  for  the  polar  subtangent,  sub* 
normal,  tangent,  normal,  and  perpendicular  to  the  tangent  of  a 
plane  curve,  when  it  is  referred  to  polar  co-ordinates. 


TANGENTS  TO   I'OLAR   CURVES. 


153 


Let  AB  be  the  curve,  Q  the  pole,  F  the 
point  to  be  referred,  QX  the  fixed  axis 
from  which  the  variable  angle  PQX  is 
retkoned,  QP  the  radius  vector,  TQN  a 
line  drawn  through  the  pole  Q^  perpen- 
dicular to  the  radius  vector  PQ^  and  limit- 
ed by  the  tangent  PT.  and  the  normal  PiV", 
QS  a  perpendicular  on  the  tangent  fi'om 
the  pole.  Then  QT  is  called  the  polar 
subtangent,  and  QN  the  polar  subnormal. 

Put  QP=  r,     angle  PQX  =  6,     angle  QPT  =  w, 

angle  PT^X  =  i,     QD  =  x,     DP  =  y. 

Then     QT  =z  QP .  tan  QPT  =r  r  tan  w  =  r  .  tan(i  —  &) 
tan  i  —  tan  & 


=  r . 


1  -f-  tan  i  tan  ^ 


But 


dy 

tan  I  z=^  ~-r, 
ax 


tan  ^ 


tani^  = 


dy 
dx 


1  + 


a;    dx 


Now  if  we  change  the  independent  variable  from  x   to  d,  we 

must  employ  the  formula  ~  —  —^ 

dd 


t&nu  = 


d^ 
d^ 


dx 
dd 


dx         dy'' 


(!)• 


And  from  the  formula  for  passing  from  rectangular  to  polar  co-or- 
dinates, we  have  x  =z  r  cos  d,  y  =^  r  sin  ^,  which  being  differentiated 
with  respect  to  ^,  observing  that  r  is  a  function  of  ^,  we  get 


dx       dr  ^ 

-7-  =  —  •  cos  5  —  r  .  sm 

aJ       d^ 


dy       dr  .     .    ,  . 

-Vf  =  -xr^iv.  ^  +  r  cos  &. 
dd       d6 


154  DIFFERENTIAL    CALCULUS, 

and  these  substituted  in   (1)  give 

r  cos  ^l-rr'  sin  &  -\-  r  cos  ^  |  —  r  sin  ^(  -- 
\dd  I  \d& 


cos  fl  —  r  sni 


tan  w  = 


rcosdl— -cosfl  —  r  sin  ^j-|-  r  sin  ^j— 'sind  +  ^cosdj 


dr~     dr 

dd 

.  • .  subtangent  ^7^=  r  tan  u  = 
Also  subnormal  V-^  ^  ~ 


dr  dr 

dd 


QT~  d6 
Tangent  FT  =z  y^QF^  +  QT^  =  ^ y  1  + 


f^  — . 
dr^ 


Normal  FN  =  ^/QF^  +  QN^  =  \/r^  + 
FQ  X  QT  r2 


dr^ 

dd^' 


Perpendicular   QS 


FT 


EXAMPLES. 


V    ^  s^ 


116.  1.  The  spiral 
of  Archimedes  whose 
equation  is  r  =  a&. 


dd-  "''     rf7"  a' 


dd         r' 

subtan  QT  =  r^  -7-=  — ,      subnorm  ON  =  ^  =  a. 
dr        a^  ^  d6         ^ 


tan  FT 


-V^- 


norm  PiV=  yVH^  perp  §^ 


^•Ha'* 


TANGENTS  TO   POLAR  CURVES. 


156 


2.  The  logarithmic  spiral  r  -.:=.  a  . 

In  this  curve,  the  numerical  value  d  of  the  arc  which  measures 
the  variable  angle  is  the  lo^rarithm  of  the  value  of  the  radius  vector  r, 
in  the  system  whose  base  is  a. 

-77  =  loff  aa  =ir .  loff  a.  .  * .    Subtan  = 

d^  ^  ^  log  a 


=  mr,  where 


m  =  modulus.     Subnormal  =  r .  log  a  =  — • 


This  curve  cuts  every 
radius  vector  under  the 
same  angle  ;  that  is,  the 
tangent  at  any  point  is 
inclined  to  the  radius  vec- 
tor  at  that  point  in  a 

constant  angle. 

d^ 

J  or      tan  u  =.r-j- 

dr 


1 


=»?. 


r  log  a     log  a 

If  a  =  e  the  Naperian  base,  then  log  a  =  1,  tan  w  =  1  and  u  =  45°^ 
and  QT=  QN=r. 

3.  The  lemniscata  of  Bernouilli,  r^  =  o?  cos  21 

dv  —  T^  —o^ 

r-z-=:— a2sin21      .*.   subtan  rr——.——r,   subnorm  =  •  sin 21 

dd  a2sni2^'  r 


perp 


vVM-«*sin2  2^    V^a4cos22a  +  a4sin22^       <^^ 

This  curve  has  the  form  of  the  figure 

8,  is  perpendicular  to  the  axis  AB  at 

A  and  B^  and  forms  angles  of  45*  with   ^\ 

AB  at  the  pole  Q.     For  when  ^=0, 

dr 
OT6=if,r  —  a,  and  — -  =  0.     And  when  6  =  45°,  or  135°,  or  225**, 

or  315°,  then  r  =  0. 


d& 


156  DIFFERENTIAL  CALCULUS. 

Rpctilinear  Asymptotes. 

117.  A  rectilinear  asymptote  to  a  curve  is  a  line  which  touches 
the  curve  at  a  point  infinitely  distant  from  the  origin,  and  yet  passes 
■within  a  finite  distance  of  the  origin.  ^ 

118.  If  in  the  difiTerential  equation  of  a  tangent  line 

y  —  y^-=.  — ^  {x  —  ar^),  we  make  successively  ar=0,  and  y=0 
we  shall  obtain  for  the  distances  intercepted  on  the  axes, 

Now  if  when  either  x-^  or  y^  becomes  infinite,  one  or  both  of  these 
values  should  prove  finite,  the  curve  will  have  an  asymptote  whose 
position  will  be  determined  by  the  values  of  ;r'  and  ?/'. 

Tf  ^'  —  a,  and  y'  z=.  h  when  a  and  h  are  both  finite,  the  asymptote 
will  cut  both  axes:  if  ^'  =a  and  y'  =  oo ,  the  asymptote  will  be 
parallel  to  the  axis  of  y\  and,  finally,  if  ar'  =  oo  and  y'  =  6,  the 
asymptote  will  be  parallel  to  the  axis  of  x. 

119.  When  the  curve  is  referred  to  polar  co-ordinates,  there  will 
be  an  asymptote  M'henever  the  subtangent  (which  is  then  equal  to 
the  perpendicular  from  the  pole  upon  the  tangent)  becomes  finite 
for  an  infinite  value  of  the  radius  vector.  Its  position  will  be  fixed 
also,  since  it  will  be  parallel  to  the  radius  vector;  that  is,  it  will  form 
w^ith  the  radius  vector  an  indefinitely  small  angle.  The  existence  of 
an  asymptote  may  be  ascertained  from  the  equation  of  the  curve  by 
findinsf  what  value  of  &  will  render  r  infinite.     If  the  same  value  o/ 

o 

6  makes  r'^  —  either  finite  or  zero,  there  will  be  an  asymptote  parallel 
•  dr 

to  the  radius  vector,  and  passing  through  the  extremity  of  the  sub. 
tangent. 

120.  1.  The  hyperbola  aV  —  ^^^^  =  — »^^^- 


cAr,     ahj^     a''   0   M      y^^         a  V         y^^       a 


RECTILINEAR  ASYMPTOTES.  157 


.  .  b      . fjx,      I        a?      hx-,      . 

Also    y,= — 'x,/x,'^—a^=i — -  \/  1 ;,= — '  when  a;,  or  y,  =  oo. 

•'^      a  ^    ^  a    \         x-^^       a 

dy-,        hx,        hx, 
^        ^^         ^  dx^         a  a 

.  ,  dx^  bx^     a  - 

and  x'  =  X.  —  y,    ~  =z  X. •  -—  =  a;,  —  ar,  =  0. 

d(/^         ^        a       h 

.  • .  The  hyperbola  has  an  asymptote  passing  through  the  origin, 
and  forming  with  the  axis  of  a;  an  angle  whose  tangent  =  ±.  — 

2.  The  logarithmic  curve  y  :=a''. 

dy       ,  ,  dx,  a*i 

-J-  =  log  a .  a',  x'  =  X,  —  y.  --^  =  x,  — =  a:,  —  m. 

,  dy,  x.a^\ 

Now  when  x^=.  -foo,yi=  +  oo,  .•.a;'  =  oo  and  y'  ^=z  cc  and 
the  corresponding  tangent  is  not  an  asymptote.  ~~ 

But  when  ajj  =  —  oo ,  y^  =  0.  .  • .  a;'  =  —  od  and  y'  z=i  0,  and 
therefore  the  axis  of  x  is  an  asymptote. 

x^ 

3.  The  cissoid  whose  equation  is  y'^  = or  ^Zry'^—y'^x—x^=(S 

d/V  ~~  X  — 

•*^  "t"         o 


y.'' 


.  • .  a;'  n:  2r,     when     x-^  =  2r     and     y^  =  cc  . 

Also  y'  =  y^_..^J^?^=y^_^i(5!r^) 

—  QC  when  a?i=2r. 


4y— 2a:j 
• .  The  cissoid  has  an  asymptote  parallel  to  y,  at  a  distance  2r 


from  the  origin. 


r 


158  DIFFERENTIAL  CALCULUS. 

4.  The  parabola  y^— 2par. 

-f-i  =  -^    ,' .  x'  =  X.—  v.  -^-^=ar,— -2iri  =  — a;,=ao  -when  a:,  =  00. 

Also  y'=yi— a^i— Tzryj— -y3=:-?/i  =  ao  when  yi  =  QOor  iCi  =  GO, 

, ' .  The  parabola  has  no  asymptote.  • 

5.  To  find  the  equation  of  the  asymptote  to  the  curve  %fz:zax^-\-x^ 

yj  =  ao  ,      when      iPj  =  00  . 

-  +  3 


=  (a.-,^  +  V)*- 

ax^ 

a                a 

—         Ill 

■Viori        /»•     —   m 

8{o«,2+a:,3)*      s(^  +  1) 


f      3 


.  • .  y  =i  X  ■\-  -a  the  equation  of  the  asymptote, 
o 

polar  Curves.  1.  The  hyperbolic  spiral  r^  =  a. 

-—  = -.     .  • .  subtan  =7-2  -—  =  a,     for  all  values  of  r. 

.  • .  There  is  an  asymptote  which  passes  at  a  distance  a  from  the 
origin.     Also,  since  r  =  oo  when  ^  =  0,  the  asymptote  is  parallel 
to  the  fixed  axis  from  which  ^  is  reckoned. 
2.  The  spiral  of  Archimedes  r  =  ad, 

d&       1  r2 

-7-  =  -)     subtan  =  —  =  00     when     r  =  od  , 
dr      a  a 

. ' .  The  curve  has  no  asymptote. 


CIRCULAR  ASYMPTOTES.  159 


8.  The  logarithmic  spiral       r  =  a  . 

dd       m  -  mr^ 

-r-  =  — i     subtan  = =  mr  =  go     when     r  =  oo . 

ar        r  r 

.  • .  There  is  no  asymptote. 

4.  The  Lituus  r^  =  a. 

iU  2a2  2a2/-2       2^2       ^        ^ 

3-  = 7-     .  • .  subtan  =  — p—  =  —  =  0     when  r  =  00 

ar  r**  r-^  r 

Also   r  =z  CD    when  ^  =  0.     .  • .  The  fixed  axis  is  an  asymptote. 


Circular  Asymptotes. 

121.  When  the  equation  of  the  curve  has  such  a  form  as  will  ren- 
der r  =  Si  finite  value  when  ^  =  00 ,  the  curve  will  make  an  infinite 
number  of  revolutions  about  the  pole  before  becoming  tangent  to  a 
circle  whose  radius  =  a.  This  circle  is  therefore  called  a  circular 
asymptote.  If  r  >  a  for  every  finite  value  of  6,  the  curve  will  lie 
wholly  exterior  to  the  circle;  but  if  r  <  a  for  all  finite  values  of  &^ 
the  curve  will  lie  entirely  within  the  circle. 

1 .  Let  the  equation  be   (r^  —  ar)&'^  =  I     or     ^  = 


■y/T^  —  ar 
Then  ^  =  00  when  r  z=z  a.     And  &  is  real  when  r  >  a,  but  imagi 

nary  when  /•  <  a. 

.  • .  The  circle  with  radius  =  a  is  an  asymptote,  and  lies  within 

the  spiral. 

2.  The  curve  {ar  —  r'^)&'^=z\. 

d  z=z  =  00     when     r  z=z  a. 

'>/ar  —  r2 

Also  ^  is  real  when  r  <^a^  and  imaginary  when  r^  a, 

.  • .  The  circle  with  radius  =  a  is  an  asymptote  and  encloses  the 

curve  within  it. 


CHAPTER   II. 


CURVATURE    AND    OSCULATION    OF    PLANE    CURVES. 


122.  As  introductory  to  the  discussion  of  the  subject  of  the  cur- 
vature of  plane  curves,  the  following  proposition  will  be  found 
useful : 

Prop.  To  show  that  the  limit  to  the  ratio  of  the  chord  and  arc  of 
any  plane  curve,  when  that  arc  is 
diminished  indefinitely,  is  unity,  and 
to  deduce  an  expression  for  the 
differential  of  the  arc  of  a  plane 
curve,  in  terms  of  the  differentials 
of  the  co-ordinates. 

Let  PPj  be  an  arc  of  a  plane 
curve  APB^  whose  equation  is  y 

Put     ODz:zx,  DP=y,  I)D^=h,  i>iA=yi,  -4P=s,  ^Pi=*i. 

Then  y^  =  F(x-{~h). 

The  arc  PP^  is  intermediate  in  length  between  the  chord  PP^z=C, 
and  the  broken  line  PIP^  =  B.     If,  therefore,  we  can  prove  that 

the  limit  to  the  ratio  —  is  unity,  it  will  follow  that  the  limit  to  the 

ratio  of  the  chord  and  arc  is  unity,  and  therefore  at  that  limit  the 
expression  for  the  chord  PPj  will  be  a  suitable  expression  for  the 
arc  P7\  which  wiU  then  become  the  differential  of  s. 


Fx. 


CtJKVATURE  AND   OSCULATION   OF  PLANE  CURVES,      161 


But    -^_-P^H--Piy_V/^^-^  +  AT^  +  Pir 

and  by  dividing  numerator  and  denominator  by  h 

L    ,  dy^  r  1    </2y    ,       k        dh/   ,    ,     "1 


=^-  =  1,  when  A=0. 


V  '  "^  c^a;2  "^  r  ^  •  ^  + 173  '  5^  *  c/x3  +  ^c- 
at  the  limit. 


1  +  ^ 

arc    _  ds  _  chord  _  tan  PT _'V      ^  dx^ 

chord         '  dx~    dx     ~      dx      ~  h 


T     'V' 


£=v/^  +  &   .•...  =  ..vA^=.^+ir.3. 


Also  ds  =z  dy  -v/l  + 


dx-^ 
di^' 

123.  In  the  first  of  these  expressions  x  is  the  independent  variable ; 
m  the  second,  y. 

Cor,  If  we  wish  to  employ  some  other  quantity  t  upon  which 
*,  X  and  y  depend,  as  the  independent  variable,  we  must  use  the 
formuljB  for  changing  the  independent  variable,  viz. : 

ds  dy 

ds       dt  dy       dt 

dx  ~  dx  dx  ~~  dx 

~dt  'di 
11 


162  DIFFERENTIAL    CALCULUS. 


which,  substituted  in  the  value  of  -7-  give 

dx  ° 


V  df  ^  dfi 


124.  We  proceed  now  to  consider  the  osculation  of  plane  curves. 
Let  Y  =  Fx  (1),  and  y  =  (^x  (2)  be  the  equations  of  two  plane 

curves,  the  first  of  which  is  given  in  species,  magnitude,  and  position, 
but  the  latter  in  species  only. 

Then  the  constants  or  parameters  which  enter  into  equation  (1)  are 
fixed  and  determinate,  but  those  which  appear  in  (2)  entirely  arbi- 
trary, and  may  therefore  be  so  assumed  as  to  fulfil  as  many  inde- 
pendent conditions  as  there  are  constants  to  be  determined. 

If,  when  the  abscissa  x  is  sup- 
posed the  same  in  both  curves,  the 
condition  y  =.  Y  \s  satisfied,  the 
curves  will  have  a  common  point 
P,  but  will  usually  intersect  at 
that  point. 

If  the  condition  -f-  =  — —  be  true  also,  the  curves  will    have  a 
ax        dx 

common  tangent  such  as  SPT\  and  the  contact  is  then  said  to  be  of- 

the  first  order :  if  the  second  differential  coefficients  be  also  equal, 

d^y       d^Y 
viz.,~-  =  --— ,  the  contact  is  said  to  be  of  the  second  order;  if 
dx^       dx^ 

d^y      d^Y 

-r-r  =  —7-^,  the  contact  is  of  the  third  order,  <fec.  &c. 

dx^        dx^ 

125.  In  order  to  show  that  the  contact  will  be  more  intimate  as 
the  number  of  corresponding  equal  differential  coefficients  becomes 
greater,  let  x  take  the  arbitrary  increment  ^,  converting  y  and  Y 
into  2/1  and  Yj  respectively. 

„„  ^       ^^dY    k  ^    d-^Y    h^    ^d^Y       A3 

dx     1         dx^     1  . 2        dx^     1.2.3 


OSCULATION   OF   CURVES.  IdS 


+  &c. 


Now  the  value  of  this  difference,  which  expresses  the  distance  by 
which  the  one  curve  departs  from  the  other,  measured  on  the  line 
parallel  to  y,  will  depend,  when  h  is  small,  chiefly  upon  the  terms 
containing  the  lowest  powers  of  h. 

>  If,  then,  the  first  differential  coefficients  derived  from  the  equations 
of  three  curves  (A),  [B)  and  (6')  be  equal,  at  a  common  point,  and 
if  the  second  differential  coefficients  derived  from  the  equations  of 
(A)  and  [B)  be  also  equal,  but  those  derived  from  (A)  and  (6') 
unequal,  the  curves  (^4)  and  (B)  will  separate  more  slowly  than 
(A)  and  (C),  because  the  expression  for  the  difference  of  the  ordi 
nates  of  (A)  and  (C)  corresponding  to  the  abscissa  x  -{-  h,  will  con- 
tain a  term  including  the  second  power  of  /i,  but  the  difference  of 
the  ordinates  of  [A)  and  [B)  will  contain  no  power  of  h  lower  than 
the  third. 

126.  The  order  of  closest  possible  contact  between  one  curve 
entirely  given,  and  another  given  only  in  species,  will  depend  on  the 
number  of  arbitrary  parameters  contained  in  the  equation  of  the 
second  curve. 

Thus  a  contact  of  the  first  order  requires  two  conditions,  viz. : 

the  first  of  these  conditions  being  employed  in  giving  the  curves  a 
common  point,  and  the  second  in  giving  their  tangents  at  that  point 
a  common  direction.  Hence  there  must  be  at  least  two  arbitrary 
parameters. 


164.  DIFFERENTIAL    CALCULUS. 

A  contact  of  the  second  order  requires  three  parameters ;  one  of 
the  third  order,  four  parameters,  &c.  Hence  the  straight  line,  whose 
equation  y  z=zax  -\-  b  has  two  parameters,  a  and  6,  can  have  contact 
of  the  first  order  only. 

The  circle  (x  —  a)^  +  {y  —  b)^  z=z  r^  having  in  its  equation  three 
parameters,  can  have  contact  of  the  second  order. 

The  parabola  can  have  contact  of  the  third  order;  the  ellipse  or 
hyperbola  a  contact  of  the  fourth  order,  &c. 

The  curve  of  a  given  species,  which  has  the  most  intimate  contact 
possible  with  a  given  curve  at  a  given  point,  is  called  the  osculatory 
^urve  of  that  species. 

The  osculatory  circle  is  employed  to  measure  the  curvature  of 
plane  curves,  and  its  radius  is  called  the  radius  of  curvature  of  the 
given  curve* 

127.  Prop.  To  determine  the  radius  of  curvature  of  a  given  curve 
at  a  given  point,  and  also  the  co-ordinates  of  the  centre  of  the  oscu« 
latory  circle. 

Let  the  equation  of  the  given  curve  be  yz=:Fx  (1),  and  that  of  the 
required  circle  (x  —  aY-\-  (y  —  b)^  =z  r^  (2),  the  quantities  a,  6  and  r 
being  those  which  it  is  proposed  to  determine. 

There  being  three  disposable  parameters,  a,  6,  and  r,  in  equation 
(2),  we  can  impose  the  three  conditions 

^,  dy       dY         ^      d-^y       d^Y 
2/=i,^  =  -^     and     ^=^- 

with  which  determine  a,  6,  and  r,  and  the  contact  will  be  of  the 
second  order. 

Denote  the  first  and  second  differential  coeflicients  derived  fronn 
thr  equation  of  the  given  curve  by  p^  and  p'\  that  is,  put 


-d^  =  ^        ^"^       ■^^=^- 

Then,  since  the  corresponding  differential  coefficients  derived  from 


RADIUS   OF  CURVATURE.  165 

the  equation  of  the  osculatory  circle  must  have  the  same  values,  we 
shall  have 

Now  let  (2)  be  differentiated  twice  successively,  replacing 
I      and     gbyyand/'. 

.♦.  (ar-a)-f(y-6)/=0...(3),and  H-/2+(y_6)/'=0.. .  (4). 

The  equations  (2),  (3),  and  (4),  will  jusib  suffice  to  determine 
a,  6,  and  r. 

Thus,  from  (4)  y-b=-^ (5)  or  6=y-f— ,7 (6) 

and  from  (3)  and  (4)  a;  -  a  =  -  (y  -  b)p'  =  ^'0  +/'),  _     ^^ 

Now  combining  (2),  (5),  and  (7),  we  get 

,  _  (1  4-  p'^f      P'^l  4-  P'^Y  _  (1  -h  P'^)\ 

The  equations  (6),  (8),  and  (9),  resolve  the  problem.  To  apply 
them  to  a  particular  case,  we  form  the  differential  coefficients  p'  and 
p"  from  the  equation  of  the  given  curve,  and  substitute  their  value? 
in  (G),  (8),  and  (9). 

Cor.  Since  1  +  j:>'2  or  1  -f  ^  =  -r^,  (Art.  122)  the  value  dt 
r  may  be  written  thus 

r=±^....(10). 


166 


DIFFERENTIAL   CALCULUS. 


Remark.   We  may  omit  the  double  sign   ±  in  (9)  and  (10)  and 

regard  the  radius  of  curvature  as  an  essentially  positive  quantity  in 

all  cases.     This  double  sign  is  sometimes  employed  to  indicate  the 

direction  of  the  curvature,  being  positive  when  the  curve  present? 

its  convexity  to  the  axis  of  ar,  and  negative  in  the  contrary  case. 

But  it  seems  more  simple  to  consider  r  essentially  positive,  and  to 

cPy 
fix  the  direction  of  the  curvature  by  the  sign  of  -7-^-     It  will   now 

be  shown  that  the  sign  of  this  second  differential  will  always  be  de- 
termined by  the  direction  of  the  curvature. 

If  the  curve  be  convex 
towards  the  axis  of  x^  as 
in  Fig.  1,  and  if  an  incre- 
ment h  be  given  to  the 
abscissa  OD  =  x,  the  or- 
dinate y  will  take  an  in- 
crement 

dy   k       d^y      h^         d^y        h^ 

and  the  ordinate  of  the  tangent  will  take  a  corresponding  increment 

£T  =  -—-.-,  and  the  former  of  these  two  increments  will  be  the 
dx    I 

greater  since  the  tangent  lies  between  the  curve  and  the  axis  of  x. 


d'^y      A2        ^^y         ^3 
dJ'T^^d^'  1T2".  3 


•••^A-^^-:7l-7^  +  :7^-T-^-^  +  &c.>0. 


or  since  the  sign  of  this  series  depends,  when  h  is  small,  on  that  of 
the  first  term,  we  must  have  — '—  >  0. 

But  when  the  curve  is  concave  towards  tne  axis  of  x,  as  in  Fig.  2, 

^Pi-^7'<0,     and     .•.^|<0. 
Again,  since  the  arc  s  and  the  abscissa  x  may  always  be  supposed 


RADIUS  OF  CURVATURE. 


167 


to  increase  together,   -—  may  be  considered  as  essentially  positive, 
and  therefore  the  sign  of  r  in  (10)  would  be  controlled  by  that  of 

d^ 


It  is  in  this  way  that  the  sign  of  r  may  be  regarded  as  indi- 


cating the  direction  of  the  curvature. 


EXAMPLES. 


128.  1.  To  find  the  radius  of  curvature  of  the  common  parabola 
y2  =  2par,  at  a  given  point. 


Here 


.  r  = 


dy      p 


p  = 

dx 

'  y 

y  -. 

2    "~ 

V 

dy 
dx~ 

f" 
■/ 

0  + 

p'2)* 

(y^  + 

T?} 

or 


(normal)' 
f  — i i . 

(semi-parameter)  2 

At  the  vertex,  y  =  0,  and  .  • .    r  =  />  the  semi-parameter ;   and 
j^  =  00 ,  r  =  GO  also. 

2.  The  ellipse  ^V  _|_  ^2^2  _  ^2^2, 


P'  = 


r  - 


^2(^2y2  ^   52^2) 


B* 


.  r  = 


^AyZ  ^2y3 

r      B^xn\ 

~  A*B*- 


AY 


B^ 


At  the  extremity  of  the  transverse  axis  x  =  A  and  y=0.    .  *.  »*=— r, 

A 


and     " 


"     conjugate 


ar=Oandy  =  ^.    .-.r: 


A^ 


168 

3.  The  logarithmic  curve      y 

y      ..     1 


DIFFERENTIAL  CALCULUS. 


log a.a'  z=z 


P 


m     dx 


where  m  ■=.  modulus. 


(1  +  v'^f 


['-ST 


my 


When         y  =  0,  r  =  x  ;    and  when  y  =  co  ,  r  =z  id  also. 


4. 

The  cubical 

parabola 

y^  =  a 

2ar. 

P'-- 

o2 
-3,^' 

P" 

3./2 . 

2y     a2 
4        3y2  - 

:   T- 

2a* 

9y* 

, 

*.  y  = 

[l  4-  ^V 

L  ^vJ 

2c/4 

(%*  4-  a 

1)!. 

%^ 

When 


ao  ,  r  =  QO 


y  =  0,  r  =  00 ,  and  when  2/  = 

5.  The  cycloid,  or  curve  generated  by  the  motion  of  a  point  on 
the  circumference  of  a  circle,  while  the  circle  rolls  on  a  straight  line. 

Let  the  radius  of  the  generat- 
ing circle  =  a.  Place  the  origin 
at  F,  the  vertex  of  the  cycloid. 
Put  VD  =  X,  DP  =  y,  the 
point  F  being  that  which  de- 
scribes the  curve  AP  VB,  while 
the  circle   rolls  on  the  line  ACB. 

Then   PD  =  BF  i-  FP  =  DF  -f  EG  since   EP  and    CF  are 
parallel.     Also,  since  each  point  of  the  semi-circnmference  CFV  has 
been  in  contact  with  the  semi-base  CA  we  must  have  arc  CFV=  CA 
and  similarly  arc  FP  =  FA  =  arc  CF. 
.  • .  By  subtraction 

CA-JEAz=CFV-CF  or  CE  =  FV  ]  and  .'.  PB  =DF-^FV 


RADIUS   OF  CURVATURE.  169 


X 

But        DF  =  v2ax  —  x'\     and      i^F  =  a  versin-^  — 


Hence  the  equation  of  the  cycloid  is 


y  =  ^'Xax—o?  +  a .  versin-^ 


a 
1 


y^       ^-^      ^  «  /2a 


-y/2ax  —  ic2  /a;      x^  ~  V 

V     a~^ 


i'"  =  - 


Xy/%IX  —  x"^ 


(2«)'*-v/tiaa:  —  a;^ 


=  2  -v/2a(2a  -  a:), 
a  yx 


'^/^ax 


x" 
or,  r  =  2  chord  PE. 

129.  Prop.  At  the  points  of  greatest  and  least  curvature  of  any 

curve,  the  oscula^ory  circle  has  contact  of  the  third  order. 

The  condition  which  chai-acterises  these  points,  is  that  the  differen- 

dr 
tial  coefficient  —   shall  reduce  to  zero,  since  r  is  a*  minimum  when 
dx 

the  curvature  is  greatest,  and  a  maximum  when  it  is  least. 

But   by    the   general   formula   for   the   radius   of  curvature, 

(1  4-  P""}         ,  ,  .       d'y 

t  =  —Y''  ^'^  ^^''^'  ^^  P"""'°  dx"  ^  ^    '  » 

ZL  -  Z =:  0. 

dx  jp"2 

^      1+/2  v*;- 

d^y 
This  is  the  value  of  the  third  differential  -— ,   at   the   points  of 

greatest  and  least  curvature,  of  any  curve;  and  if  it  can  be  shown 


170  DIFFERENTIAL   CALCULUS. 

that  the  third  differential  coefficient  in  the  osculatory  circle  has  the 
same  value,  it  will  follow  that  the  contact  must  be  of  the  third 
order. 

But  in  the  circle  we  have  already  found  y  —  b  = rf—i 

which  being  identical  with  (1),  the  contact  must  be  of  the  third 
order. 

130.  Prop.  If  two  curves  have  contact  of  an  even  order,  they 
will  intersect  at  the  point  of  contact ;  but  if  the  order  of  their  con- 
tact be  odd,  they  will  not  intersect  at  that  point. 

If  Z"  =  Fx^  and  y  =  (pa:,  be  the  equations  of  the  two  curvesj  the 
difference  of  their  ordinates  corresponding  to  the  abscissa  x-\-  h, 
will  be  expressed  by 

'       '''~\dx       dx]\    \   )^\vlx-^       dx'').l.% 

Now  when  the  order  of  contact  is  even,  the  first  term  of  this  dif- 
ference which  does  not  reduce  to  zero,  must  contain  an  odd  power 
of  ±  A,  and  must  therefore  change  sign  with  A,  thus  imparting  a 
change  of  sign  to   Y^  —  y^,  in  passing  through  the  point  x,y. 

Hence  the  first  curve  will  lie  alternately  above  and  below  the 
second,  intersecting  it  at  the  point  x^y. 

But  if  the  order  of  contact  be  odd,  the  first  term  in  the  difference 
will  contain  an  even  power  of  ±  A,  which  will  not  change  sign  with 
/i,  and  therefore  there  will  be  no  intersection ;  the  first  curve  lying 
entirely  above  or  entirely  below  the  second. 

Cor.  The  osculatory  circle  intersects  the  curve,  except  at  the 
points  of  greatest  and  least  curvature. 

For  usually,  the  circle  has  contact  of  the  second  order — but  at  the 


RADIUS  OF  OCRVATURE.  171 

points  of  greatest  and  least  curvature,  the  contact  is  of  the  third 
order. 

Cor.  At  those  points  of  ^y 

a  curve  where  jt?"  =  0,  a     


straight  line  may  have  con- 
tact of  the   second   order, 

and  it  will  intersect   the  curve.     If  p'"  =  0,  also  there  will   be  no  ^ 
intersection  unless  jp""  =  0,  also. 

131.  Prop.  To  find  a  formula  for  the  radius  of  curvature,  when 
any  quantity  i,  other  than  the  abscissa  x,  is  taken  as  the  independent 
variable. 

To  effect  this  object,  we  must  substitute  in  the  value  of  r,  already 

found,  the  values  of  p'  =—^  and  p"=  y^,  given  by  the  formula 
for  changing  the  independent  variable,  viz.  : 

dy       dt 
dx  ~  dx^ 
U 
We  thus  obtain 

"^  d^         \d3^  dJ^J     ^d^"dt~  di^'di 


d?y 

d'^y 
df 

dx 
'  dt 

d'^x 
df'' 

dy 
'  dt 

dx' 
dt^ 

r  = 


d'^y  ld.c'^\^         *  dx' 


\di^  "^  d^)  dt? 


d^y    dx        d'^x    dy       d'^y   dx       d'^x    dy 
Ji?  "di  ~  d^  "di       dfi  '  di  ~"di^  "cTi 

dx       dx 
Cor.  If  X  be  the  independent  variable,  —  =  —  =  1, 

dP 

d'^x  dx' 

tnd     T^  =  0,      .  • .   y  =  -;— ,  the  common  formula. 
dt^  ay 

d^ 


(1) 


172  DIFFERENTIAL   CALCULUS. 

diJ      dv  d!^v 

If  y  be  the  independendent  variable,  —  =  — -  =  1,  and  —  =  0. 

_ds^ 


dy'^ 


If  s  be  the  mdependent  variable,  —  =z—-z=.  1, 

•'•   '"'^0?^   dx _d^    df ^^'' 

dd'^    ds       di^     ds 

djtP"       0??/^       tt.s^ 
But  -—  +  -^  =  -^^  =  1,  which,  being  differentiated  with  respect 

to  5,  gives 

^    d^x       dj_    d^_f.  /ox 

ds '  d,^  "^  ds '  ds'  -  ^>  •  •  •  •  v^;- 


(4). 


1 

dy 
ds 

d-x    dx" 

~ 

d'x' 

ds'    ds'' 

d''x    dy 

ds' 

dy 

ds''    ds 

ds 

dx 

and  similarly 

ds 

•  •  •   (5). 

ds' 

And,  finally,  by  squaring  the  equation  (2),  and  adding  to  the 
denominator  of  the  second  member,  the  square  of  (3),  which  is  equal 
to  zero,  there  results,  by  reduction, 

r''  =  — -: — r„ — ;.     and 


132.  Prop.  To  obtain  a  formula  for  the  radius  of  curvature  of 
curves  when  referred  to  polar  co-ordinatos. 


RADIUS  OF  CURVATURE.  173 

Adopting  the  variable  angle  ^  as  the  independent  variable,  denoting 

iho  radius  vector  by  r,  and  the  radius  of  curvature  by  i2,  we  have, 

from  the  formulae  for  the  transformation  of  co-ordinates, 

dx  dr 

X  z=ir  cos  ^,  2/  =  ^  sin  ^,  .  • .  —j-  =  —  r  sin^  -f-  cos  4  — , 

^^=rcos&  H-  sm^-r-- 
dd  dd 


r  COS  ^  —  2 

sin  ^  -—  +  COS  & 
dd 

d^r 

dd^' 

dhj 
d6^-       ' 

rsin4  +  2i 

cos  6  —  +  sin  & 

d^r 
dd^ 

dr 
rut     —-  =  »i  and 
dS      ^^ 

d^r 
dd^=P' 

and  substitute 

in  the  general  valu 

of  the  radius  of  curvature. 

rdx^ 

d^y    dx 
c^2    ^d 

dy^i 

dd-'. 

d'^x   dy 
dd^    d& 

[r2-f  jt?i2(sV4-f  cos2-a)f 

{r'+P^ 

.^)* 

j^ 

r^  -+-  2py^—rp2       r^  +  2pi^  —  rp2 
where  H  is  the  polar  normal. 


EXAMPLES. 


133.  1.  The  logarithmic  Spiral  r  =  a  . 

i?i  =  log  a .  a^=z  -^,   p^  =  log2a .  /=  ^ 

P  _         JV^         _    jsr^m^    _  jisr  (7-2  +  ^v^)  _  «. 

.  • .  The  radius  of  curvature  of  the  logarithmic  spiral  is  always 
equal  to  the  polar  normal. 


174  DIFFERENTIAL  CALCULUS. 

2.  The  spiral  of  Archimedes  r  =  a&. 


Pi  =  a,    pr^  =  0  .'.  R 


r2  +  2a?- 


When  r 


0,  E  =  -  a,  and  when  r  =  oo  ,  i2  =  r 


3.  The  hyperbolic  spiral  r&  =  a. 


Pi  = 


"P"  -  ■"  T'  ^^'' 


2a 


2r3 
a2 


'.  R  = 


+  2--  — 


a3 


a"         a* 

v  hen  r  =  0,  i2  =  0,  and  when  r  ==  od  ,  i?  =  oo . 
4.  The  litims  rH  =  a^. 


I'l 


.•.i2  = 

When  d  =  0,  r  =  00  and 
Ji=cc  ;  when  ^  =  1,  /=«, 
125 


1 

"~2 

,.6 

»-3 

r(4a*  4-  r*)* 

r2  + 

~4a* 

3r6    ~ 
4  a* 

•  2a2(4a*  -  r*) 

3r« 
4a* 


and  R—a- 


6 


when 


r = a -Y/2^or  r* = 4a*,72  =  GO 

and  when  ^  =  00  r  =  0  and  i2  =  0. 

134.  A  curve  may  be  characterized  by  an  equation  expressing  k 
relation  between  the  radius  vector  r  and  the  perpendicular  p  from 
the  pole  upon  the  tangent. 

Thus  the  equation  of  the  circle  referred  to  the  co-ordinates  r  and  jt) 
IS  r  =p^  the  pole  being  at  the  centre.  That  of  the  logarithmic 
spiral  is  r  =  cp,  &c. 


RADIUS  OF  CURVATURE.  175 

135.  Prop.  To  obtain  a  formula  for  the  radius  of  curvature  of 
curves  referred  to  the  radius  vector  and  the  perpendicular  upon  the 
tangent. 

From  the  general  value  of  the  perpendicular  when  the  ctirve  is 
referred  to  the  ordinary  polar  co-ordinates  r  and  ^,  viz. :   (Art.  115.) 


P=      , ==:     weobtam     -rr^  = -^  -  »-"  =  ;>i 


V         ^   c/c)2 


which,  differentiated  with  respect  to  d,  gives 

dr    dP-r       ^r"^    dr         2r*    dp      ^    c?^ 
Yd"d^-'^~^'~d^'~^'~dd~    ''!&' 

r.  ■,     .      .       r.     dp  .         .      dp    dr       1  T   .T    ,      ^dr 
Substituting  for  -^  its  value  —  •  —  and  divide  by  2  -^r' 

d^r  _  2r3        r*    dp  _ 

"*•  ~d6i'~'^'~"^"di-~'^'~^^' 

dr  d^r 

Now  substituting  the  values  of—  and  -—  in  that  of  i?,  we  get 

ij  =      (^  +^1  ) ^         p^         ' 

r2  4-  2pi^  —  rp^ 


r  dr 

~  dp~    dp 
Tr 


^«(7-)-'(f-?|-') 


Ex.  The  involute  of  the  circle  whose  equation  referred  to  p  and  > 


is  p^  =  r^ 


dr       p  r»         ^^         P  n 2 

dp        r  dp  r  ^ 


CHAPTER  in. 


EVOLUTES    AND    INVOLUTES. 


136.  The  curve  which  is  the  locus  of  the  centres  of  all  the  oscu- 
latory  circles  applied  to  every  point  of  a  given  curve,  is  called  the 
evolute  of  that  curve,  the  latter  being  termed  the  involute  of  the 
former. 

137.  Prop.  To  determine  the  evolute  of  a  given  curve  y  =  Fx. 

If  in  the  formulae  for  the  co-ordinates  of  the  centre  of  the  oscu- 
latory  circle,  viz. :    (Art.  127.) 


P 


,^+P" 


(1)     and     h  =  y  -\- 


l-fp'2 


(2), 


p  p 

we  substitute  the  values  of  p'  and  p'\  derived  from  the  equation  of 
,the  curve  y  =z  Fx  (3),  we  shall  have  the  three  equations  (1),  (2), 
and  (3),  involving  the  four  variable  quantities  x,  y,  a,  and  6;  and  by 
eliminating  x  and  y  the  result  will  be  a  general  relation  between 
a  and  6,  the  coordinates  of  the  required  evolute.  This  equation 
being  independent  of  x  and  y  will  apply  to  every  point  in  the 
desired  curve. 

138.  In  most  cases  the  necessary  elimina- 
tion is  quite  difficult;  the  following  are  com 
paratively  simple  examples. 

1 .  The  evolute  of  the  common  parabola. 

Here  we  have     y^  _  ^px  ....  (1). 

.•.w'=:^-i     and     «"  =  _^. 


EVOLUTES  AND  INVOLUTES. 


y       y^      p^  p 


177 
(2). 


6  =  y  — 


+  p2     y^ 


y^ 


.  r 


y2  p2  ^         p2  ^  p2 


(3). 


4      i 

From  (2)  and  (3)  we  get    x  =z  — -—■>     and     y^  —p  b  ', 


and  these  values  substituted  in  (1)  give 


p^b^  =:'ilp 


-P 


•••^'=^.(«-^)^ 


3  'Zip 

the  equation  of  the  semi-cubical  parabola,  whose  axis  coincides  with 
that  of  the  given  curve ;    the  distance  Aa  between  the  vertices 
being  =  p  the  semi-parameter. 
2.  The  ellipse 

A^y^  ^  B^x"^  =  A^B^  ....  (1). 

B^x       „  J5* 

A^f  -f  B^x"^ 

2 


1   -f  p'2  = 


< 

> 

ry 

"^ 

^ 

d 

or     a  = 


xA^{A^B^-  Ahj-^)-  B*x^  _  x^P 


and     i=:y- 


A^B^  =-ir    where  /«  =  ^2  ^^a, 

?/(^V  +  ^'•^-)       AY—B^if(A'^B'^-BH^)  y^P 


A'^B^ 


f   I 


b*B^ 


£* 


.    .  a;^  =:■ 


which  values  substituted  in  (1)  give 


A^ 


+  I1^~  =  A^B^, 


r 


12 


c^ 


y 


a 


V 


178 


DIFFERENTIAL    CALCULUS 


0- 


the  equation  of  the  required  evolute. 

When      a  .—  0,     6  =  ±  ^ ;     and  when     5  =  0,  a  =  ±  ^. 

The  curve  consists  of  four  branches  presenting  their  convexitiesi 
towards  the  axis,  and  tangent  to  each  other  as  shown  in  the  diagram. 
The  equilateral  hyperbola  referred  to  its  asymptotes. 


xi/  =z  (?• 


(1). 


/  =  -^'  /' 


X^  X' 


c*  +  x^     p' 


c'*  4-  a;'^      ,  ,   c*  4-  a;* 

r,a  —  x-\ PT^— 5     h  —  y-\- 


2x^ 

3 


2c2^ 


and 


\/a-^b-^^a  -b  = 


2x3  /c 


.(1), 


III 
Hence  by  multiplication  (a  +  b)    —  (a  —b)    =  (4c)  . 

139.  Prop.  Normals  to  the  involute  are  tangents  to  the  evolute, 
From  the  equation  of  the  osculatory  circle  (x —a)^-^  (y—by^r*, 
we  get  by  differentiation 

x-a+p\y-b)  =  0 

a  relation  alike  applicable  to  the  y 
circle  and  the  given  curve,  since 
X,  y  and  p'  are  the  same  in  both. 
Now  wheL  v»  e  pass  from  a  point 
x,y  to  another  point  on  the  circle, 
the  quantities  x,  y  and  p'  must  be 


EVOLUTES  AND  INVOLUTES. 


179 


considered  variable,  but  a  and  b  constant ;  but  when  we  pass  to  a 
point  on  the  curve,  x^  y,  p\  a,  and  6  will  all  vary,  and  in  both  cases 
p"  will  be  the  same. 
The  first  supposition  gives,  by  differentiating  (1)  with  respect  to  ar, 

i+/2+y'(y-6)  =  o (2), 

and  the  second  gives 


1 


^'  fy^-/^+y'(y-6)=o 


dx  dx 

"\^ence  by  combining  (2)  and  (3) 


(3> 


dx       db 
'  da  ~~  da  "^ 


db 


Now  —  represents  the  tangent  of  the  angle  formed  by  the  axis 

of  ar,  with  the  tangent  to  the  evolute  AB  at  the  point  P^,  and 

J-  =:  tangent  of  the  angle  formed  by  the  same  axis  with  the  normal 

PPi  to  the  involute  LM  at  the  point  a:,y;  which  normal  passes 
through  the  point  P^.  Hence  this  normal  not  only  passes  through 
the  point  cr,i,  but  it  also  coincides  in  direction  with  the  tangent  to 
the  evolute  at  that  point. 

140.  Prop.  The  difference  of  any 
two  radii  of  curvature  is  equal  to  the 
arc  of  the  evolute  intercepted  between 
those  radii. 

Resuming  the  equation 

(x  -  a)2  +  (y  -  6)2  =  r2, 

and  differentiating  with  respect  to  a, 
us  an  independent  variable,  we  obtain 


180  DIFFERENTIAL    CALCULUS. 

(.-„)g-,)+(.-.)g-g=4. 

Bat      (._„)|  +  (,-.)|  =  |[._„+(,_.)g].0. 


,    .  ,.dh  dr 


db             \        y  —  b  ,  ,/,       </<^2v  dr 

or  since  —  = ,= ,     .-  .{x  —  a)\\  -{- —-\z=— r--"(\), 

da  p'      X  —  a  ^  ^\         d(i?l  da     ^ 

(db'^\ 


or, 

Dividing  (1)  by  (2),  there  results, 


(■-S)'= 


_  dr 
da 


But      1 1  +  -7"ol  =  "Tj  where  s  is  the  arc  of  the  evolute  which 
\         da^J        da  .  . 

terminates  at  the  point  a,  b. 

.'.    —-=z:jp-—.     and     ds  =  :=pdr. 
da  da 

Thus  it  appears  that  the  increment  of  s  is  always  numerically 
equal  to  the  increment  of  r. 

Hence  s  must  always  differ  from  r  by  a  constant  quantity,  or  we 
must  have  s  ■=:  c  =f  v,  and  similarly  for  the  arc  «j,  which  terminates 
at  the  point  a^^b^,  s-^  =  c  ^  r^,  .  * .  s^  —  s  =  r  —  r^,  which  result 
agrees  with  the  enunciation. 

141.  In  finding  the  evolutes  of  polar  curves,  it  is  usually  most 
convenient  to  employ  the  relation  between  r  and  p,  the  radius  vect 
and  the  perpendicular  on  the  tangent;  thus,  let  r  =  radius  vecto. 
the  given  curve,  p  =  the  perpendicular  on  the  tangent,  r^  =  radiu 
vector  of  the  evolute,  p^  =  the  perpendicular  on  its  tangent. 


EVOLUTES  AND   INVOLUTES. 


181 


Then  since  the  radius  of  curvature 
PPy^  =  JK,  at  the  point  P,  is  tangent 
to  the  e volute  at  Pj,  the  perpend  icu- 
ular  QT^^  is  parallel  to  the  tangent   ^ 
TP. 

Also  QT  is  parallel  to  PP^ 

.-.  PT^:=QT^p, 
and  PT=  QT^^p^. 

.',    ri2=:  B'-{-r^-2Bp,  .  .  .  (1). 

dr 


(2). 


Also     P  =  r 


dj) 


(3)  (Art.  135).     And     r  ^  Fp, 


(4),  th« 


equation  of  the  given  curve. 

By  eliminating  r,  p,  and  P,  between  (1),  (2),  (3),  and  (4),  there 
will  result  a  relation  between  rj  and  p^  which  will  be  the  equation 
of  the  required  evolute. 

Ex.  The  l(»garithmic  or  equiangular  spiral  r  =  cp (4). 


-—=ze,     and     R  —  cr^  . 
dp 


(3).     r-^  =.  p-^  +  p^\, (2). 


ri2  =  722  ^  r2  -  2P/? (i). 

From  (1)  and  (3),  r^  =  cV^  -f-  r^  _  ^crp,  which  combined  with 
(4)  gives 

r2  -  c2^2(i  4.  c2)  _  2c>2  -  c^.pi{c^  __  1) (5). 

From  (2)  and  (4),  c^p^  =^2  ^ ^^^2^    ^r,    p^c^-V)  =  p^\  .  .  (6). 

Then    from  (5)  and  (6),     rj^  =  c^pi^     or,     rj  ==  cjOj,    the  equdr 
tion  of  a  similar  and  equal  spiral. 


CHAPTER   IV. 


CONSECUTIVE    LINES    AND    CURVB8. 


142.  If  different  values  be  successively  assigned  to  the  constants 
or  parameters  which  enter  into  the  equation  of  any  curve,  the 
several  relations  thus  produced  will  represent  as  many  distinct 
curves,  differing  from  each  other  in  form,  or  in  position,  or  in  both 
these  particulars,  but  all  belonging  to  the  same  class  or  family  of 
curves.  When  the  parameters  are  supposed  to  vary  by  indefinitely 
small  increments,  the  curves  are  said  to  be  consecutive. 

Thus  let  F(x,y,a)=  0,  .  .  .  .  (1),  be  the  equation  of  a  curve, 
and  let  the  parameter  a  take  an  increment  h,  converting  (1),  into 
F(x,i/,a  -\-  h)  =  0,  ....(2),  then  if  h  be  supposed  indefinitely 
email,  the  curves  (1)  and  (2)  will  be  consecutive.  Moreover,  the 
curves  (1)  and  (2)  will  usually  intersect,  and  the  positions  of  the 
points  of  intersection  will  vary  with  the  value  of  h,  becoming  fixed 
and  determinate  when  the  curves  are  consecutive. 

143.  Prop.  To  determine  the  points  of  intersection  of  consecutive 
lines  or  curves. 

To  effect  this  object,  we  must  combine  the  equations 

F(x, y, a)  =  0, (1).     and     F(x, y,a  +  h)  =  0,  .  ..  ,  (2). 

and  then  make  h  =  0,  in  the  result. 

Expanding  (2)  as  a  function  of  a  -f-  ^^  by  Taylor's  Theorem,  and 
observing  that  x,  and  y,  being  the  same  in   (1)  and   (2),   (since  they 


CONSECUTIVE  LINES  AND  CURVES.  183 

refer  to  the  points  of  intersection.)  are  to  be  considered  constant  in 
this  development,  we  obtain 

dF(x,y,a)    h 


F{x,y,a  +  h)  =  F{x,y,a)  + 


da  1 


_^d2F(x^y^J^ 
^         da?  1.2^ 

But  F{x,  y,  a)  =  0, 

.     dF{x,  y,  a)    h       d^F{x,  y,  a)     h?      ,    .      __  ^ 
'    '  Ja         '  1  ^         da^        *  1  .  2  "^  ^^-  -  "* 

,.  .,.       ,      ,      dF(x,y,a)  ,   d^F(x,y,a)     A      .    , 

And  when  h=0  this  reduces  to     — ^L^l^  =  0 (3). 

da  ^ 

The  two  conditions  (1)  and  (3),  serve  to  determine  the  co-ordi. 
nates  x  and  y,  of  the  required  points  of  intersection. 

,144.  Fx.  To  determine  the  points  of  intersection  of  consecutive 
normals  to  any  plane  curve. 

The  general  equation  of  a  normal  is 

(y-yi)/  +  ^-^i  =  o, (1). 

in  which  x^^  y^,  and  p\  are  parameters,  all  of  which  vary  together. 
Differentiating  (I)  with  respect  to  x-^^  and  observing  that  y^  and 
p'  are  functions  of  arj,  and  that  x  and  y  are  to  be  considered  con- 
stant, we  get 

(3'-y.K-p'"-i=o,....(2)- 

or,     ,  =  ,.  4-  i±/-V  .  (3).  and  .  • .  .  =  ..-'^/^.  ■  ■  i^)- 

The  values  (3)  and  (4)  being  identical  with  those  of  the  co-ordi- 
nates of  the  centre  of  the  oscuhitory  circle,  it  follows  that  consecu- 
tive normals  intersect  at  the  centre  of  curvature.  This  principle  is 
sometimes  employed  in  determining  the  value  of  the  radius  of 
curvature. 


184  DIFFERENTIAL    CALCULUS. 

'  '  145.  Prop.  The  curve  which  is  the  locus  of  all  the  points  of  in. 
tcrsection  of  a  series  of  consecutive  curves  touches  each  curve  in 
the  series. 

If  we  eliminate  the  parameter  a  between  the  two  equations 

F(^x,y,a)  =  0  ...  (1)     and     ili'-lllll  =  0  ...  (2), 

the  resulting  equation  will  be  a  relation  between  the  general  co-ordi- 
nates X  and  y  of  the  points  of  intefsecti>»n,  independent  of  the  par- 
ticular curve  whose  parameter  is  a,  or,  in  other  words,  the  equation 
of  the  locus. 

Resolving  (2)  with  respect  to  a  the  result  may  be  written 

and  this  substituted  in  (1)  gives 

^hy,<p(^,y)]  =  o (3), 

which  will  be  the  equation  of  the  locus. 

dy 
Now  if  the  differential  coefficient  -7^   be   the   same  whether  de- 

dx 

rived  from  (1)  or  (3),  the  two  curves  will  have  a  common  tangent 

at  the  point  x,y,  and  therefore  will  be  tangent  to  each  other. 

Differentiating  with  respect  to  a:,  we  obtain  fvam.  ( I ) 

dF{x,y,n)    ,  dF(x,y,a)    dy 


dx  dy  dx 


=  0  .  .  .  .  (4), 


a„d  from  (3),  ''^T^. ?/. P(^..'/)1  +  dF[.,yM-,y)]  _  dy^ 
^  '  dx  dy  dx 


d:plx,y)  L     dx    J  ^  ^ 


But  the  first  and  second  terms  of  (4)  and  (5)  are  identical,  and  the 

third  term  of  (5)  is  equal  to  zero  by  (2). 

dv 
Hence  the  values  of  ~  given  by  (4)  and  (5),  and  by  (1)  and  (3), 


ENVELOPES. 


185 


are  the  same,  and  consequently  the  two  curves  (1)  and  (3)  arc  tan- 
gent to  each  other. 

146.  The  curve  (3)  which  touches  each  curve  of  the  series,  is 
called  the  envelope  of  the  series. 

147.  1.  To  determine  the  envelope  of  a  series  of  equal  circles 
whose  centres  lie  in  the  same  straight  line. 

Assuming  the  line  of  centres  as  the  axis  of  a:,  the  equation  of  one 
of  these  circles  will  be  of  the  form 


{X 


+  f 


»-2  =  0. 


(1)- 


in  which   a   is  the   only  variable 
parameter. 

Differentiating  with  respect  to 

a,  we  get 

-  2a;  +  2a  =  0  . 


t^//.,. 


f( 


.  v-^4 


(a) 


From  (2)  a  —  x,  and  this  substituted    in  (1)  gives 


=  0. 


y 


±r. 


This  is  the  equation  of  two  straight  lines  parallel  to  and  equidis- 
tant from  the  axis  of  ar,  a  result  easily  foreseen. 

2.  The  envelope  of  a  series  of  equal  circles  whose  centres  lie  in 
the  circumference  of  a  sriven  circle. 


Let 


+  yi' 


0....(1) 


be  the  equation  of  the  fixed  circle.  / 

that  of  one  of  the  moveable  circles. 

The  variable  parameters  are  x-^  and  y^^  the  latter  being  a  function 


of  the  former. 


dy\ 


From  (2)  we  have     ~  ^{x  —  x^  —2{y  —  y^) -~^  z=  0  .  ,  .  (3). 

OiX't 

But  from  (1)     a:^  +  yi  — '  =  0     or     "^^^  -       ^i  -  ^i 


dxy 


dxi 


Vi 


V'V^--V 


IB^ 


DIFFERENTIAL  CALCULUS. 


This  value  in  (3)  gives 


-  (^  -  ^i)  +  (y  --vA?  -  a^i^) 


V^ 


=  0. 


or 


x-i- 


y^i 


=  0,     and 


(4). 


Now  combining  (1),  (2),  and  (4),  so  as  to  eliminate  rCj  and  yi,  we  get 


©r 


^  +  y +  ^1  ==  T^' 

y/x^  +  y^ 

.  • .  ^x^  +  2/2  -  rj  =  ±  r.        .  • .  a;2  4-  y2  _  (^^  ±  y)2. 

This   is   the   equation    of  two   concentric   circles  whose   radii   aie 
r^  4-  y  and  r^  —  r  respectively. 

3.  The  curve  which  touches  every  chord  connecting  the  extremi 
ties  of  conjugate  diameters  of  an  ellipse. 

Let  Q^Py^  and  Q2P2  ^®  conjugate  diameters  of  the  ellipse  ACBD^ 
Xy  and    y^  the   co-ordinates  of  P^,  c 

X2  and    1/2  those  of  Pj. 

Put    AO  =  a,     OC-b, 

tan  PjO^  =  tan  ^1  =  —  =  /], 

tan  Pg^^  =  tan  ^2  =  ^  =  /g- 
^2 

Then,  since  by  the  property  of  the   ellipse,  ^j/j  =  ~ 


.  —  h'^x-^X2  =  o.'^yiy2    and     X2 


62.f, 


Also  aV^2  4    J2^^2  ^  a2y^2  4.  J2^^2  ^  ^2^32  4.  ^|l^ 


fl'^y2'^ 


62Xi2 


(a2yj2  +  ^,23.^2) 


or 


ENVELOPES.  187 

.•.^_^  =  i     and    .•..,  =  ^    and    .,  =  — ^  =  -  ^ 

■  • 

.  * .  The  equation  of  the  line  P1P2  i^ 

'^.(3/  +  ^)-2/^(--f)-a^  =  0....(l). 
Differentiating  (1)  with  respect  to  x^  we  get 

But  aVi^  +  ^'^1'  =  aW (3),    and    .  •.  ^  =  _  ^ 

Hence  (2)  can  be  reduced  to 

.         aV.(y  +  g  +  6^.,(.-f)=0....(4). 
Combining  (1)  and  (4)  we  have         x^  =  ^^^^T^^S" 

-nrl  •     ,/  -  -bx^i^x  -ay)  __  _   gfe^  (6.g  -  ay) 

'    '  Vi-      a{bx-j-ay)      ~       2  (aV  +  ^^•^■') 

These  values  reduce  (3)  to  the  form 

aH^  [(bx  +  ayf  +  (bx  -  qy)^]  _  ^     _^       jr^^  __ 

4  (aV  +  62a;2)2  "    '     ^^     FT  "^  K     ~    ' 

/T         a 

the  equation  of  an  ellipse  whose  semi-axes  are  ci\/ -  and  b\/-.  and 

which  is,  therefore,  similar  to  the  original  ellipse. 

4.  The  envelope  of  a  series  of  lines  drawn  from  every  point  in  a 
parabola,  and  forming  with  the  tangent  angles  equal  to  those  included 
between  the  tangent  and  the  axis. 


188 


DIFFERENTIAL   CALCULUS. 


Let  PD  be  one  of*  the  lines. 

Put            DPT  z=  PTD  z=:&,AE  =z  a;,  EP  =  y,. 
Then  PDE  =2^,  and  the  equation  of  the  line  PD  is 
y  —  yi  =  tan^r2^(^  —  x^) (1). 

_              ^^,       2tan^  dx,         ,   .  ^     ^        dih      p 

But  tans  2^  = — 2I—  "T  2  >  ^"^  ®"''^*^ 2/i  =rPa-;, 


1— tan^J 


«^-^]     yi' 


.  tan  2^  = 


.^1 


i;??/i 


;^2      Vi'-P^ 


"t  yyx^  —  fy  +  />Vi  —  2/)x.y,  =  0.  .  .  . 

Differentiating  ('i)  with  respect  to  (/j,  we  find 


(2). 


2////i  +  /^^  —  2/>a;  =:  0,     and     i/j  z=z 
This  vahie,  substituted  in  (2),  gives 
4r>2^2_4^3j._^^4  ^2px  —  rP- 


—  2/?a; 


22/      ' 

2;?a?  —  jo2 


=  0, 


or  by  reduction     (2ar  —  pj-  +  (2//)2  =  0  .  .  .  (3). 

This  can  be  satisfied  only  by  making  2jr  — ;?  =  0  and  y  =  0, 

.  • .  (3)  represents  a  point  whose  co-ordinates  are  x  =  -p  and  y  =  0. 

Thus  the  lines  will  all  pass  through  the  focus;  as  might  have  been 
foreseen  from  the  well-known  property  of  the  pai-abola. 

5.  From  every  point  in  the  circumference  of  a  circle,  pairs  of 
tangents  are  diavvn  to  another  circle  ;  to  find  the  curve  which  touches 
every  chord  connecting  correspond  in  jj  points  of  contact. 


ENVELOPES 


189 


Let  Pi  be  a  point  on  the  first 
circle  P1P2  and  P-^P^  a  pair  of  tan- 
gents, P2P2  <^*i®  <^f  the  chords,  0 
the  origin  at  the  centre  of  the  se- 
cond circle,  x-^y^  the  co-ordinates  of 
P|,  ^^2^2  those  of  Pg'  ^3^3  those  of 


Then  tj  —  y^  —  — ^  (a;  —  2:2) (1)  is  the  equation  of  the 


chord  P^Py 


Also   ?/jy2+^i^2=^^ (2)  the  equation  of  the  tangent  P^Pi 

applied  to  the  point  P^. 

y^y^-^rx^x^—r"^  ...  (3)  the  equation  of  the  tangent  P3P1  applied  to 
the  point  Pj. 


Then     ^i  (2/2  -  ^s)  +  a^i  (arg  —  iCg)  =  0,    and 

X 

which  reduces  (1)  to     y^y^^ ^-  {x  —  x^) 


-Vz 


y\ 


or 


yVx  +  xx^  =  ViVz  +  ^1^2  =  '•^ 


(4). 


Now  differentiating  (4)  with  respect  to  arj,  we  get 


dx. 


-{■  x  :=0,     But  2/i2  4-  (arj—  a)2  =  ^1= 


(5). 


a  =  0     and 


y 


y\ 


li+a:=:0, 


or  ya:j  —xy^  —  ay (6). 

Combining  (4)  and  (6)  we  have 

2/1  =  -^-- — ^      and      iTj  =  — —- — f-. 
'^^         a;2  +  y^  ^        x^  -\-  y^ 

These  values  substituted  in  (5)  give 

(rc2  +  ?/2^2  \  i  / 

Hence  the  curve  required  is  always  a  conic  section.  It  is  a  circle 
when  a  =  0,  an  ellipse  when  a  <  rj,  a  parabola  when  a  •=.  Vy  and  a 
hyperbola  when  a  >  Vy. 


CHAPTER    V, 


SINGULAR    POINTS    OF    CURVES. 


148.  Those  points  of  a  curve  which  enjoy  some  property  not 
common  to  the  other  points,  are  called  singular  points.  Such  are 
multiple  points,  or  those  through  which  several  branches  of  the  curve 
pass;  conjugate^  or  isolated  points;  cusi^s^  or  points  at  which  two 
tangential  branches  terminate ;  points  of  inflexion,  &c.  These  will 
be  examined  successively. 

Multiple  Points, 

149.  These  are  of  two  kinds,  viz. :  1st.  When  two  or  more 
branches  intersect  in  passing  through  a  point,  their  several  tangents 
at  that  point  being  inclined  to  each  other ;  and  2d.  When  the 
branches  are  tangent  to  each  other,  their  rectilinear  tangents  being 
coincident. 

150.  Prop.  To  determine  whether  a  given  curve  has  multiple 
points  of  the  first  species. 

At  such  a  point,  there  must  be  as  many  rectilinear  tangents,  and 

uv 
therefore  as  many  different  values  of  the  differential  coefficient  — 

as  there  are  intersecting  branches. 

Let  F(x,y)  =  0  =  «,.....  (1),  be  the  equation  of  the  given 
curve,  freed  from  radicals. 


MULTIPLE   POINTS.  191 

du 

__  du  rfjB 

Then   since  p'  =  -^  z=z —,  and  since  differentiation  never  in- 

dx  du 

dy 
troduces  radicals  where  they  do  not  exist  in  the  expression  differen- 
tiated, the   value  of  'p'  above  given   cannot  contain  radicals,  and 
therefore  cannot  be  susceptible  of  several  values,  unless  it  assumes 

the  indeterminate  form  -• 

Hence  the  condition  p'  z=i-  will  characterize  the  points  sought. 

To  discover  whether  such  points  exist,  and  if  so,  to  find  their  posi- 

tions,  we  form  the  partial  differential  coefficients  -7-  and  -r  from 
^  dx  dy 

the  equation  of  the  curve,  then  place  their  values  equal  to  zero,  and 

determine  the  corresponding  values  of  x  and  y. 

If  these  values  prove  real,  and  satisfy   (1),  they  may  belong  to  a 

multiple  point.     We  then  determine  the  value  of />'  by  the  method 

applicable  to  functions  which  assume  the  indeterminate  form  -,  and 

if  there  be  several  real  and  unequal  values  of  jt:}',  they  will  corre- 
spond  to  as  many  intersecting  branches  of  the  curve,  passing  through 
the  point  examined. 


EXAMPLES. 

151.   1.  To  determine  whether  the  curve  x^  -j-  Saar^y  —  ay^  =  0, 
has  multiple  points  of  the  first  species. 

|3  =  a,*  -f  2aa;2y  —  ay3  =  0,  .  .  .  .  (1). 

du  du 

_  =,  4x3  -f  4«^y^ (2).      —  =  2aa;2  -  Zay\ (3). 


_  \x^  4-  ^oxy 
^   ^3ay2  — 2aa-? ^  *' 


192 


DIFFEKENTIAL  CALCULUS. 


Placing  (2)  and  (3)  equal  to  zero,  we  get 

xix^  +  «y)  =  0, (5). 

and,  2;r2  _  3^2  ^  o (6). 

Combining   (5)  and  (G)  we  have   three 
pairs  of  values  for  x  and  y,  viz. : 

a;  =  0,     and     y  =  0, 

OT,x=-\-  -a-^,  and  y  —  —  -a,  or,  x=z  —  ~a^/Q,  andy=  — -u. 

The  first  pair  of  values  will  alone  satisfy  (1),  and  therefore  the 
origin  is  the  only  point  to  be  examined. 

Placing  X  =:  0,  and  y  =  0,  in  (4),  there  results 


^      12a;2  4-  4ay  +  4a xp' 


0 


Qui/p'  —  4.ax 


when 


or  by  substituting  for  numerator  and  denominator  their  differential 
coefficients, 

24a;  +  Sap'  +  4axp"  _       Sap'  __     ^^^^^     j  a;  =:  0 

—      w   P.n  ^^ 


p^: 


.X  = 


6«y^  +  (yat/p"  —  4a       ijap'^  —  4a 
. '.    p'(Qap'^  —  4a)  —  Saj)',  and  consequently 
p'  =  0,     or,  p'  =  -\-  \/2,     or,  p'  =  —  -y^. 

Hence  the  origin  is  a  triple  point,  the  branches  being  inclined  to 
the  axis  in  angles  whose  tangents  are  0,   +  y^,  and  —  y^, 
pectively. 

The  form  of  the  curve  is  shown  in  the  diagram. 

0  =  w (I). 


res. 


2.  The  curve  ay^  —  x^y  —  ax^ 
^"^       -  3a;2y  _  3aa;2  =  0, (2).    ^  =  3ay2  -  a;3  =  0 (3). 


dx 


From  (2)  and  (3),  a;  =  0,  and  y  =  0,  or,  a:  =  a -^,  and  y  :=—a. 
The  first  pair  of  values  satisfies  (I),  but  the  second  does  not. 
Therefore  the  origin  is  the  point  to  be  examined. 


MULTIPLE  POINTS.  193 

Hence  p'  =    ,/ 7 —.,-  =  -^. r^-o =  a  ^^^^^  \  a 

Gy  4-  ]  ^xp'  -f  3a:2yo"  +  6a        6a 


Ottp'-^  H-  i5aj/p"  —  ijx  iSap'"^ 

.  • .    p'3  __  1^     and    p'  =  1. 


when 


j  a;  =  0 


This  being  the  only  real  value  of  />',  there  is  but  one  branch 
passing  through  the  origin,  and  therefore  the  curve  has  no  multiple 
points. 

3.  The  cyrve  x*  —  2ay^  -  Sahj^-—  2a'^x^  +  a*  =  0  =  -w (1). 

—  =  4:{x^  -  a'^x)  =  0 (2).     ^  =  -  6(ay2  +  a^^)  =  0.  .  .  (3). 

\JLJO  (jv  il 

From  (2)  and  (3)  we  get  six  pairs  of  values,  viz. : 

a:  =  0,  and  y  =  0,  or,  ic  =  0,  and  y  =  —  a, 
or,  X  —  a,  and  y  =  0,  or,  x  =  —  a,  and  y  =  0, 
or,        a;  =:  a,     and     y  =  —  a,     or,     a;  =  —a,     and     y  =  —  a. 

But  of  these  six  pairs  of  values,  the  2d,  3d,  and  4th,  are  the  only- 
ones  which  satisfy  (1),  and  therefore  there  are  but  three  points  to 
be  examined. 

'  _  ^'^^  —  ^<^^^  _      6a;2  —  2a^      _    4  (  a;  =  dc  a 

^  ~  Say'  +  3.a2y  ""  {ijay  -\-  Sa^)p'  ~  Sp'     ^  ^"       [y  =  0 

and  »'  =r  — —      when      \     ~ 

^i>  \y  —  —a- 

.  • .  p'  =  ±  I  -  J    at  the  point  where  x  z=:  a  and  y  =  0 


''  =  '© 


"  "      a;  =  —  a  and  y  =  0 

;?'  =  ±  /^  P  «  "      a;  =  0  and  y  =  -  a. 

Thus  the  curve  has  three  double  pomts. 
13 


194  DIFFERENTIAL   CALCULUS. 

152.  Proj).  To  determine  whether  a  given  curve  has  multipk 
points  of  the  second  species. 

Here  the  mode  of  proceeding  is  similar  to  that  in  the  last  propcr- 
sition,  but  the  resulting  values  of  jt?'  prove  equal  although  given  by 
an  equation  of  the  second  or  higher  degree. 

Ex.  The  curve     x'^  +  x'^y'^  —  ^ax^y  -f-  a^y"^  =  0  =  w (1). 

~  =4a:3+ 23^2/2 _i2aa:y=i0  .  .  (2),   ~—'lx^y—Ux^^'i.a^y=^  . .  (3). 

GLJu  CLU 

From  (2)  and  (3)  a:  =  0  and  y  =  0,  and  this  is  the  only  pair  of 
values  which  will  satisfy  (1).  Hence  the  origin  is  the  only  point  to 
be  examined. 

,_  \2axy—2xy'^—^x^  _  12ay+\2axp'—2y^—4xyp'  —  12x^_    0 


2x^y  —  ijax'^-\-'2a^y  4xy-\-2x^p'  —  l2ax-^2a^p'  2a  ^p" 

when    X  =  0  and  y  =  0.     .'.  p'"^  =  ——  =  0     and    »'  =  ±0. 

2a^ 

And  the  origin  is  a  double  point  of  the  2d  species. 

153.   We  may  prove   directly  that  at  a  double  point  of  the  2cl 

kind,  the  condition  p'  =  7:  is  always  fulfilled. 

Thus  suppose  the  two  branches  to  have  contact  of  the  w'*  order. 
Then  the  first  n  difftrential  coefficients  will  be  the  same  for  the  two 
branches,  but  the  («  -f-  1)^^  differential  coefficient  will  be  different 
at  the  double  point. 

Let  P~-  +  Q  —  0 (1)  be  the  result  obtained  by  differen- 
tiating the  given  equation  once,  in  which  F  and  Q  are  functions  of 
r  and  y,  the  original  equation  having  been  freed  from  radicals. 

By  repeating  the  differentiation  n  times,  we  get 


CONJUGATE   OR   ISOLATED   POINTS.  195 

in  which  P  is  the  same  as  m  (1),  and  ^j  is  a  function  of  ir,y,and 
the  differential  coefficients  of  the  several  orders  less  than  {n  -f-  1). 

Now  the  {n  -\-  X)th  differential  coefficient  has,  by  supposition,  two 
different  values  a  and  h  for  the  same  values  of  P  and  Q^ 

,'.  Pa+  Qi  =  0,     and     Pb -^  Q^  z=  0, 

and  by  subtraction  P{a  —  b)  =  0.     .'.  P  =  0  since  a  and  b  are 
unequal. 

This  value  of  P  substituted  in  (1)  gives  ^  =  0. 

'   '  dx  P      0 

Multiple  points  of  the  2d  species  are  characterized  by  having  but 

one  value  (or  rather  two  or  more  equal  values)  for  -^,  but  several 

d'^y 
unequal  values  for  -~  or  some  higher  coefficient. 


Conjugate  or  Isolated  Points. 

154.  These  are  points  whose  co-ordinates  satisfy  the  equation  of 
a  curve,  but  from  which  no  branches  proceed.  When  p'  assumes 
the  imaginary  form  for  real  values  of  a;  and  y,  the  corresponding 
point  will  be  isolated,  as  the  curve  will  then  have  no  direction ;  and 
since  imaginary  values  occur  only  where  radicals  are  introduced,  the 

condition  p'  z=  -  will  also  hold  true  in  such  cases. 

The  converse  proposition,  viz. :  that  at  a  conjugate  point  p'  will 
be  imaginary,  is  not  always  true ;  for  if  in  the  development 

any  one  of  the  differential  coefficients  should  prove  imaginary,  y, 
would  be  imaginary  also. 


196  DIFFERENTIAL  CALCULUS. 

To  determine  with  certainty  whether  a  point  (rt,6)  is  isolated,  sub- 
stitute successively  a  +  h  and  a  -^  h  for  x,  and  if  both  values  of  y^ 
prove  imaginary  (A  being  small),  the  point  will  be  imaginary; 
otherwise  it  will  not. 

155.  If  the  coefficient  7)'  =  —  be  found  to  have  multiple  values, 

«ome  being  real  and  some  imaginary,  we  may  regard  the  result  ks 
indicating  the  indefinitely  near  approach  of  a  conjugate  point  to  a 
'•eal  branch  of  the  curve. 

EXAMPLES. 

156.  1.  To  determine  whether  the  curve 

«y2  _  a;3  -f  4ax2  -  ba^x  +  Sa^  =  0  =  «  .  .  .  .   (1) 
has  conjugate  points. 

^  =  -  3a:2  +  8a^  -  5a2  =  0 (2),     ~=2ay  =  0 (3). 

w,x  dy 

5 

From  (2)  and  (3),     x  =  a  and  y  =  0,  or  x  =z  -a  and  y  =  0. 

o 

The  first  pair  of  values  satisfies  (1),  and  therefore  the  point  {afi) 
must  bo  examined.  .    ; 


,       3x2       g^a-  _^  5^2       6a;  -  8a  1 

y  =  — 


^  —      .^     /  ;     when      \ 

2ay  'Zap'  p'  \y  —  0. 

.•./2=  _  1,  p'  ^  ±y=~r. 


This  result  being  imaginary,  we  conclude  that  the  point  examined 
is  isolated. 

2.  The  curve  {c^y  —  x^f  z=z{x  —  a)^  (x  —  bf^  in  which  a  >  6. 
u  ~.  (c^y  -  a:3)2  ^  (x  -  af  (x -^  b)^  =  0 (I), 

^  =  2c^c^y-x^)=0 (3), 

—  =:-6a:2(c2y-a;3)-.5(a;--a)*(a;-6)6-G(ar~a)»(a:~6)«=0  .  .  .  (2>. 


CONJUGATE    POINTS.  19? 

■Th«  equations  (2)  and  (3)  give 

X  =ia  and  y  =  — ,  or  a;  =  6  and  y  =  -j ; 

both  of  which  pairs  of  values  satisfy  (1),  and  therefore  both  require 
examination. 

_Qx^(c^y-x^)       5(ar  -  a)^(^  -  by  +  6(a;  -  af{x  -  by 

=         4.  J^ 2-1 L^^ L.^ i-  =  -^  when  ar=:5, 

3a2 
=:^-whena;i=<z. 

Thus/)'  is  real  at  both  points.     But  if  we  substitute  b  ±  h  for  4 
hi  (1),  and  solve  with  respect  to  y,  we  get 

both  of  which  values  of  y  are  imaginary  when  k  is  taken  less  that 

a  —  6 ;  so  that  the  point  where  x=:b  and  yrr  —  is  a  conjugate  point, 
although  p'  is  real. 

This  result  is  confirmed  by  forming  the  succeeding  differential 
coefficients ;  thus 

p"=:l^Qx  +  ^.^(x-a)^{x-bY-\-\6{x-a)\x-bf 

+  Q{x—a)   (x—b)  I  =  — ,  when  x  z=k 
This  is  a  real  value  also.  • 

But  the  next  coefficient  will  contain  the  term  G{x'-a)  —Q[b—a) 
which  is  imaginary,  since  a"^  b. 

The  value  x  =:z  a  does  not  belong  to  a  conjugate  point,  as  is  .seen 


m 


DIFFERENTIAL   CALCULUS. 


by  substituting  a  ±  h  for  x  in  (1),  and  solving  with  respect  to  jf, 
^us, 

which  is  real  when  A  >  0,  but  imaginary  when  A  <  0. 


Cusps. 


157.  A  cusp  is  that  peculiar  kind  of  double  point  of  the  second 
•jpecies  at  which  two  tangential  branches  terminate  without  passing 
fhrough  the  point. 

C^usps  are  of  two  kinds,  viz. : 

1st.  That  in  which  the  two     ^ 
branches  lie  on  different  sides 
of  the  tangent,  as  in  Fig.  1. 

2d.  That  in  which   they  lie 

on  the  same  side  of  the  tangent,  as  in  Fig.  2. 

dif 
The  test  of  a  cusp  is  that  ~~  shall  have  two  real  and  equal  values 

at  some  point,  (a.i),  and    that   when  we  substitute  a-\-h  and  a — h 

for  X,  we  shall  find,  in  one  case,  two  real  and 

unequal  values  of  y,  and  in  the  other   two      "^ 

imaginary   values.     The  only  exception    to 

this   is   that  offered   by  the  case  shown   in 

Fig.  3,  where  a  cusp  of  the  first  kind  occurs 

u 
at  a  point  P,  with  the  tangent  parallel  to  the 

Rxis  of  y.     It  will  then  be  more  convenient  to  form  the  value  of 

dx 

-r-,  which  should  be  ±  0,  and  to  try  whether  the  successive  substi- 

dt/ 

tution  of  b  -{-  h  and  b  —  h  for  y  will  render  a?,  in  one  case,  real  and 
double,  and  iri  the  other  imaginary.  The  condition  ;?'  =  -  serves  as 
A  guide  in  selecting  the  pdints  to  be  examined. 


CUSPS.  199 


EXAMPLES. 

158.   1.  To  detennine  whether   the  curve  (5y  —  ca;)^  =  («  —  a)* 
has  a  cusp,  and  if  so,  of  which  kind. 

u-{hy-  cxf  -{x  —  afz^^ (1), 

^  =  -  2c(6y  -  ex)  _  5(a:  -  a)*  =  0 (2). 

^  =  26(6y-c.:)  =  0 (3). 

dC 

From  (2)  and  (3)  we  obtain  a:  =  a,  and  y  = —^ 
and  as  these  values  satisfy  (1),  we  must  examine  the  point  (a,  -t-) 

,     2ciby  -cx)-\-^ix-aY      0        .  ixz=a 

p'~—S-l — —-'— — ^ ^=1-     when       )  m 

,       ^  2b{by  -^cx)  0  (^  =  ~r 

2bcp'—2c^-\-20(x-aY      2bcp'-2c^ 
= 2by-.^c =  2b^^-^^2fc^    ^^^"     ^  =  ^' 

.  • .  by^  -  2bcp'--c\     p'2-2-^p'=  -  -^  andp'  =  -^±  0. 

'  .  • .  There  are  two  equal  values  of  jt?',  and  consequently  two  tan 

tt.C 

gential  branches  proceed  from  the  point,  a,  — • 

Now  put  successively  x  ==  a  -{-  h,  and  x  z=z  a  —  h,  and  solve  with 
respect  to  y. 

when  x=a-{-h,  y=z  — ~j^  ^j  ^wo  real  and  unequal  values. 


when  x=a—h,  y=i ~-^ ~  two  imaginary  values. 

cic 
Hence  there  is  a  cusp  at  the  point  a,  — ,  and  the  tangent  at  thai 

point  is  inclined  to  the  axes  of  x  and  y. 

Again,  the  ordinate  V  of  the  tangent  corresponding  to  the  abscissa 

.    7    •    "^    .      /  T        ac  -{-  ch     .  .  .    .  ,  ^    , 

a  +  /i,  is  -y-  -tp/i  = which  is  greater  than  one  of  the  cor- 


200 


DIFFERENTIAL   CALCULUS. 


responding  values  of  y,  and  less  than  the  other.  Therefore  the 
branches  lie  on  different  sides  of  the  tangent,  and  the  cusp  is  of  the 
jfirst  kind. 

Remark.  The  kind  of  cusp  can  usually  be  found  \qyy  easily  by 
examining  the  values  of  the  second  diiferential  coefficient;  for  the 
deflection  of  the  curve  from  the  tangent  is  controlled  by  the  sign  of 

— ^«     Hence,  when  the  two  values  of  this  coefficient  have  contrary 

signs,  the  cusp  will  be  of  the  first  kind,  but  when  the  signs  are  alike, 
it  will  be  of  the  second  kind. 

2,  The  semi-cubical  parabola     cy'^  =  x^. 


u  —  cf—x^  .  .  .  (1), 


dn 
dx 


dn 

3x^--=0  .  .  .  (-2),    -=2cy=0...(3). 


.  •.  a;  =  0,  and   y  ==  0,  and  as  these  satisfy  (1)  there  may  be  a 
cusp  at  the  origin. 
,       3.i'2        iSx  0 

/  =  -—  =  ^r— 7  =  ^r— 7     whcn     X  =  0. 

""Zcy       2cp        2ep 


P 


—  =:  0     and    p'  =  ±0, 
"Zc 


two  real  and  equal  values. 

Now   put   0  d-  A  for  x  in   (1),  and 
there  will  result, 


when  a;  =  0  +  A,  y  = 


two  real  and  unequal  values, 


0-A,  y=±y/-- 


two  imaginary  values. 


.  • .  There  is  a  cusp  at  the  origin.  Also  the  ordinate  Y  of  the 
tangent  corresponding  to  the  abscissa  0  -f  A.  is  0  -f  p'h  =  0,  which 
being  intermediate  in  value  between  the  two  corresponding  values 
of  y,  the  cusp  is  of  the  first  kind. 

159.  Sometimes  it  is  more  convenient  to  solve  the  equation  with 
respect  to  y  before  differentiating. 


POINTS   OF   INFLEXION. 


201 


Ex. 


{y 


5)2  z=  (x-ay 


y  =  b-{-cx'  ±.{x-  a)*     p'  =  2cx±^{x--  a)*. 

Now  ij  has  but  one  value  h  4-  ca?^  or  to  speak  more  correctly,  it 
has  two  equal  values  {b  -\-  ca?  ±  0)  when  x  =  a,  and  p'  =  2ca  dt  Q, 
has  then  two  equal  values  also. 

When  x=:a  +  h,  y  =  h-\-c{a-^hy±(  +  h)  two  resil  and  unequal  values. 

"      x  —  a—h^  y  =  h-\-c{a—hy±:(—h)  two  imaginary  values. 
Hence  there  is  a  cusp  at  the  point 

(a,  b  +  ca^).  ^ 

Also  p"  z=2c  ±f-l{x  -  a)^  =  2c  ±  0 

when     X  ^=z  a. 

And  since  the  two  values  of  p"  have  the 

same  sign,  the  cusp  at  the  point  (a,  b  4-  cd^)  is  of  the  second  kind. 

The  kind  of  cusp  would  also  appear  by  comparing  the  ordinate  Yof 

the  tangent  with  the  two  values  of  y. 

For  when  x  =  a  -^  h,   Y  =z  b  -\-  ca^  -\-  p'h  =  b  -{■  ca^  -{•  2cah, 
which  is  less  than  either  value  of  y,  when  h  is  small. 


Points  of  Inflexion. 

160.  Points  of  inflexion  or  contrary  flexure  are  those  at  which 
the  curve  changes  the  direction  of  its  curvature,  being  successively 
convex  and  concave  towai-ds  a  fixed  line  as  the  axis  of  x. 

It  has  already  been  remarked  that  a  curve  is  convex  towards  the 

axis  ot  X  when  -— -  is  positive  and  concave  when  -r—  is  negative 

(JLjb  (XJC 

Hence  a  point  of  inflexion  will  be  characterized  by  having  the  second 
diflTerentiai  coefficient  aff*ected  with  contrary  signs,  at  points  situated 


202  DIFFEEENTIAL    CALCULUS. 

near  to,  but  on  different  sides  of  the  point  in  question.     But  since  a 
variable  quantity  changes  its  sign  only  when  its  value  passes  through 

zero  or  Infinity,  the  condition  — -|-  =  0  or  — -  =  oo  will  belong  to 

a  point  of  inflexion.     But  the  converse  is  not  necessarily  true,  for 

the  sign  of  —    does  not  always   change  after  its  value  has  reached 

0  or  00  .     We  must  therefore  see  whether  a  change  in  the  sigQ  of 

dP'y 

-r—  will  or  will  not  occur. 

We  may  also  recognize  a  point  of  inflexion  by  the  consideration 
that  at  such  a  point  the  tangent  intersects  the  curve,  and  therefore 
the  ordinate  of  the  tangent  will,  on  one  side  of  the  point  be  greater, 
and  on  the  other  less  than  the  corresponding  ordinate  of  the  curve.  • 


EXAMPLES. 

161.   1.  The  cubical  parabola  d^y  =  x^, 

y  =  — '     p  =  — ;r-'    i'    =  -^  =  0     when     a?  =  0 
a^  o/'  a^ 

' .  The  origin  is  a  point  to  be  examined. 

Put     X  z=:  0  -\-  h,  and  y  =  y^, 

X  =z  0  —  h,  and  y  =  ^g* 

Then  5^.->0,  _- 


--r  \''K 


dx''   ~        d^  ^ 

Hence  the  origin  is  a  point  of  inflexion.     The  condition  p"  =  oo 
gives  X  z=:  oD ,  apd  therefore  is  not  applicable. 

162.  Sometimes  it  happens  that  two  of  the  peculiarities  which 
chaiacterize  singular  points  occur  at  the  same  point  of  a  curve. 


POINTS  OF  INFLEXION.  203 

Ex,  a?y^  -  2abx^y  —  a;5  =  0  =  w  .  .  .  .   (1), 

Y 


^ 


du 

^=  -Adbxy-bx^  =0 (2), 

^  =  2a3y-2a6a:2  =  0...  .  (3). 

The  equations  (1),  (2),  and  (3),      _ 
are  all  satisfied  by  the  values 
a;  =  0,  y  =  0. 

Aahxy  -f  5a;*       0        ,  r  a:  =  0 

and  there  is  either  a  cusp  or  a  double  point  at  the  origin,  the  axis 
of  X  being  tangent  to  the  curve. 


IfiCr=0  +  A,     y  —  -—  ±\/ ,  two   real  values,   ore 

greater  and  the  other  less  than  the  ordinate  (0)  of  the  tangent. 


\^  X  —  0  —  h^     y  =z~^  ^\/ 4 '  ^^^  ^^^^  values  when 

h  is  small,  but  both  greater  than  0. 

Hence  there  is  a  double  point  of  the  second  species  at  the  origin, 
jand  one  branch  of  the  curve  has  an  inflexion  at  that  point. 

163.  In  addition  to  the  singular  points  already  described,  two 
Other  classes  may  be  noticed,  viz.  :  Stop  Points,  or  those  at  \vl\ich  a 
«ingle  branch  terminates  abruptly ;  and  Shooting  Points,  at  which 
two  or  more  branches  terminate  without  being  tangent  to  each 
other.  Both  are  of  rare  occurrence,  but  the  following  are  examples 
of  curves  belonging  lo  these  classes. 

1.  y  ■=  X  Ao^x.  This  curve  has  a  stop  point  at  the  origin. 
For,  y  has  but  one,  value,  and  that  is  real  vshen  a;  >  0  ;  but  the 


20^ 


DIFFERENTIAL    CALCULUS. 


value  of  //  is  impossiMe  when  a:  <  0,  since  negative  quantities  can 
not  properly  be  regarded  as  having  any  logarithms. 

2.     !/  —  ^  tan""^  -,     or,     y  =  ^  cof^a;. 

X 

This  curve  has  a  shooing  point  at  the 

origin,  for 


dx 


1  X 

tan-i--— -— 
X       i  ■\-  x^ 


=  tan-i(-j-  00  ) 


-^  =  1.5708     when     a:  =  +  0 

it 


=  tan-^(-  00  )=—-*=  -  1.5708     when     x  =  -  0, 
and  whether  x  be  positive  or  negative,  y  will  have  but  one  value. 

164.  When  a  curve  has  the  spiral  form,  and  is  therefore  more  c,  i- 
veniently  referred  to  polar  co-ordinates,  we  may  distinguish  tJ  e 
existence   of    a   point   of  contrary   flexure  by    the  condition    that 

—  =  0  at  tliat  point,  and  that  it  shall  have  contrary  signs  on  differ 
ar 

ent  sides  of  that  point.     This  we  proceed  to  show. 


Q  Q 

In  Fig.  1,  the  curve  is  concave  to  the  pole  Q ;  and  in  Fig.  2,  it  is 
convex. 

la  the  first  case  r  and  p  increase  together^  and  therefore  -j~  is  posi. 

tive.     In  the  second  case,  p  diminishes  as  r  increases,  and  therefore 
dp 


~-  IS  negative.      Hence,  in  passing  through  a  point  of  contrary 


dt) 
flexure,  ~-  will  change  its  sign,  becoming  equal  to  zero  at  that  point, 

pjr   -^  plainly  could  not  b<»come  infinite,  since  j9  cannot  exceed  r. 


CHAPTER   VI. 


CURVILINEAR    ASYMPTOTES. 


165.  When  two  curves  continually  approach  each  other,  and  meet 
vnly  at  an  infinite  distance,  each  is  said  to  be  an  asymptote  to  the 
other. 

166.  Prop.  To  determine  the  conditions  necessary  to  render  two 
curves  asymptotes  to  each  other. 

Let  the  curves  be  referred  to  rec- 
tangular axes,  and  let  the  ordinates 
iJP  and  EP\  corresponding  to  the 
same  abscissa  OE  =  x^  be  express- 
ed by  means  of  the  equations  of 
thfc  curves  in  terms  of  x.  The 
difference  PP'  =  y^  —  y  can  then  be  expressed  in  terms  of  x,  and 
if  tbiv  difference  be  reduced  to  zero  by.  making  a;  =  oo ,  (being 
finite  for  all  other  values  of  x,)  the  curves  will  be  asymptotes  to 
each  other. 

This  condition  is  fulfilled  only  when  the  difference  (expanded  into 
a  series,  contains  none  but  negative  powers  of  a;,  without  an  abso- 
lute term,  for  in  such  cases  only  will  the  difference  Vi  —  y  become 
zero  when  x  =:  cc  . 

Hence  we  must  be  able  to  express  yx~  V  in  the  form 
y^  —  y  =  Ax-<^  -f  Bx-^  4-  Car-"  +  &c., 
or  the  difference  x^  —  x  of  the  two  abscissae,  corresponding  to  the 
same  ordinate,  must  admit  of  being  expressed  in  the  form 
Xi  —  x  =  A^y-^x  +  B^y-K  -f-  C^y^x  +  &c. 


206  DIFFERENTIAL  CALCULUS. 

167.  Cor.  If  there  be  three  curves,  (^),  (^),  and  (C),  and  if 
the  difference  of  the  corresponding  ordinates  of  [A)  and  (^),  and 
that  of  the  ordinates  of  (^4)  and  (C),  be  thus  expressed. 

y2  —  yi  =  Ar-^  +  Bx-^"-^'"'  +  Cx-^''-^'''^ (1). 

2/3  -  yi  =  B,x-<^-^'^  +  Ci.r-^«+^>  +  &c,  .  .  .  .  (2). 

the  three  curves  will  be  asymptotes  to  each  other,  and,  moreover, 
the   curve   (C)   will   lie   nearer   to   i^A)  than  [B)  does.     For,  by 

making  x  sufficiently  large,  the  term  Jar-",  or  —  may  be  rendered 

greater  than  the  sum  of  the  succeeding  terms  of  (1),  or  greater  than 
the  sum  of  those  terms  increased  by  the  series  (2). 

168.  Cor.  The  curve  whose  equation  can  be  written  in  the  f(jrm 

y  =  n^Ax''  -\-  Bx^  -\-  Cx"  +  Ji.r-«i  +  B^x-^x  -|-  C^r-^^i  +  &c., 

can  have  an  infinite  number  of  curvilinear  asymptotes. 
For  by  taking  any  curve  whose  equation  is  of  the  form 

y^z=z  D  +  Ax^  +  Bx^  +  Cx""  +  ^2^"^  +  B^prK  +  &c. 

in  which  the  absolute  term  i>,  and  the  terms  involving  the  positivt* 
powers  of  a;,  are  the  same  as  in  the  given  equation,  the  difference 
l/\  ~  y  will  reduce  to  zero  when  a;  =  oo . 

169.  ProiJ.  To  find  the  general  form  of  the  expanded  value  of 
the  ordinate  in  such  curves  as  admit  of  a  rectilinear  asymptote. 

Since  the  equation  of  the  rectilinear  asymptote  has  the  form 
y  ~  AyX  -\-  j5i,  the  equation  of  the  desired  curve  must  take  the  forna 

y  =  A^x  +  ^1  +  Jar«  4-  Bx-^  +  Cxr<^  +  &c. 

170.  1.  The  common  hyperbola  a^y^  _  ^2^2  _  _  ^^2^2^ 

y  =  ±  -(a:2  _  a2)*=  dt-{x-  \a?jr-^  -\a^x-'^  -  &c.) 
IJut  y  =  zh  -  a;    is   the   equation  of  two   straight   lines   passing 


CURVILINEAR  ASYMPTOTES.  207 

through  the  origin  and  equally  inclined  to  the  axis  of  x.  Hence 
these  lines  are  asymptotes  to  the  hyperbola. 

2.  To  determine  whether  the  curve  y  z=  b{x^  — a^)  has  either 
rectilinear  or  curvilinear  asymptotes. 

By  expansion 

y  zzz  b{'jcr^  +  -a2a:-3  ^  &c.)  =  hx-^  +  -  ba?x-^  +  &c. 

But  y  =  0  is  the  equation  of  the  axis  of  x.  Hence  that  axis  is  a 
rectilinear  asymptote  to  the  curve. 

To  discover  whether  there  is  an  asymptote  parallel  to  the  axis 
of  y,  let  the  equation  be  solved  with  respect  to  x\  thus 

a;  =  dr  (a2  +  62^-2)*  =  ±  (^  .|.      h2^-\y-2  _  <^c.) 

Z 

Here  it  is  evident  that  two  lines  parallel  to  the  axis  of  y,  and  at 
distances  therefrom  equal  to  4-  a  and  —  a  respectively,  will  be 
asymptotes  to  the  curve,  their  equations  being 

X  =z  -\-  a     and     x  z=  —  a. 

The  hyperbola  whose  equation  (referred  to  its  asymptotes)  i« 
xy  =^  b  will  be  a  curvilinear  asymptote,  and  there  may  be  found  any 
number  of  other  curvilinear  asymptotes. 


CHAPTER   VII. 


TRACING    OF    CURVES. 


171.  In  this  chapter  it  is  proposed  to  give  such  general  directions 
as  are  necessary  in  tracing  a  curve  from  its  given  equation,  and  in 
discovering  the  chief  peculiarities  which  characterize  it. 

The  following  steps  will  be  found  useful : 

1st.  Having  resolved  the  equation,  if  possible  with  respect  to  y, 
let  ditTerent  positive  values  be  assigned  to  x  from  a:  =:  0  to  .c  =  <x, 
and  let  those  points  be  noticed  particularly  wh3re  y  =  0,  y  =  x,  or 
y  zzz  an  imaginary  value.  The  first  indicates  an  intersection  with 
the  axis  of  x,  the  second  shows  the  existence  of  an  infinite  branch, 
and  the  third  gives  the  limits  of  the  curve  in  the  direction  of  x 
positive. 

2d.  Aspign  to  x  all  negative  values  from  x  =  0  to  x  =  -•  oo, 
and  observe  the  same  peculiarities  with  respect  to  y  as  when  x  was 
positive.  In  both  cases  the  negative  as  well  as  the  positive  values 
of  y  must  be  examined  so  as  to  include  the  branches  below  as  well 
as  those  above  the  axis  of  x. 

3d.  Determine  whether  the  curve  has  asymptotes,  and  determine 
their  position. 

4th.  Find  the  value  of  the  differential  coefficient   -f-  and  deter- 

ax 

mine  from  thence  the  angles  at  which  the  curve  cuts  the  axes,  as 

well  as  tlie  points  at  which  the  tangent  is  parallel  to  either  axis. 

dhj 
5th.  From  the  value  of  -r-—  ascertain  the  direction  of  the  cur- 


TRACING  OF  CURVES.  209 

vature  and   the  positions  of  the  points  of  contrary  flexure  when 
they  exist. 

6th.  Determine  the  positions  and  character  of  the  other  singular 
poir.ts,  if  there  be  such. 

EXAMPLES. 

172.  1.  Let  tho  equation  of  the  proposed  curve  be 

x^  —  n^ 

Resolving  with  respect  to  y  we  have 


and  since  each  value  of  x  gives  two  values  of  y  numerically  equal 
but  having  contrary  signs,  the  curve  must  be  divided  symmetrically 
by  the  axis  of  x.  * 

\^  X  be  positive  and  numerically  less  than  a,  y  will  be  imaginary, 
and  there  will  be  no  point  of  the  curve  between  the  axis  of  y  and 
a  parallel  thereto  at  a  distance  equal  to  a  on  the  right  of  the 
origin. 

When  a:  —  a,  y  =  0,  when  a:  >  a,  y  is  real,  and  continues  so  for 
all  greaJer  values  of  .r,  becoming  infinite  when  a;  =  od  . 

If  x  be  negative  and  numerically  less  than  &,  y  is  imaginary,  and 
there  is  no  point  between  the  axis  of  y  and  a  parallel  thereto  at  the 
distance  ::=  6,  on  the  left  of  the  origin. 

When  x  =.  —h^y  becomes  infinite ;  and  when  re  <—  6,  that  is, 
negative  and  numerically  greater  than  6,  y  becomes  real  and  con- 
tinues to  increase  without  limit  as  the  numerical  value  of  x  increases, 
being  i'lfinite  when  x  z=z  —  oo . 

Th;:s  it  appears  that  the  curve  has  six  infinite  branches. 

Again,  since  x  =.  —h  makes  y  infinite,  there  is  an  asymptote 
parallel  to  the  axis  of  y,  and  at  a  di^tance  therefrom  equal  to  —  6. 

14 


210  diffp:rential  calculus. 

Also  by  resolving   the   given  equation  with   respect  to  y,  anc 
expanding,  we  get 


y  = 


-'^-4— ('-#(-4)"' 


{x  4-  by 

(1      7  Q      12  \  1 

^  ~~  o  ~  +  Q  ~i'  &;c.  I  =  db  (^  —  ^  ft  +  terms  involv 

ing  powers  of  x). 

Hence  y  __  ±i{x  —  -b)  is  the  equation  of  two  straight  lines,  which 

are  asymptotes  to  the  curve,  and   are  inclined  to  the  axis  of  x  at 
.nngles  of  45'^  and  135°  respectively. 

If  we  combine  this  equation  of  these  asymptotes  with  that  of  the 
curve,  we  shall  find  that  each  of  the  asymptotes  intersects  that 
branch  of  the  curve  which  lies  on  the  right  of  the  axis  of  y. 

Forming  the  value  of  — -  from  the  equation  of  the  curve,  we  have 
di/  2x^  4-  Sbx^  +  a^ 


dr  —  4  * 

)l{x^  -  a^Y  [x  +  b)^ 

which,  placed  equal  to  zero,  gives  the  cubic  equation 

in  M^hich  there  must  be  one  real  and  negative  root,  since  the  absolute 
term  is  positive.  The  other  two  roots  are  imaginary,  as  is  easily 
Keen  from  the  form  of  the  equation.  Thus  there  are  two  points 
corresponding  to  the  same  negative  abscissa,  one  above  and  the 
other  equally  below  the  axis  of  x.  at  which  the  tangent  is  parallel  to 
the  axis  of  x. 

By  making  ~  z=i  co  ^  we  get  ar  =  a  or  a?  =  —  6.     The  first  corres- 
ponds to  a  point  at  which  the  curve  intersects  the  axis  of  x  perpeiv 


TRACING   OF  CURVES. 


211 


dicularly.     The  second  belongs  to  the  point  of  contact  of  one  of  the 
asymptotes  as  before  seen. 

By  forming  the  value  of  -7-|,  we  should  find  that  the  curve  is  con- 
cave to  the  axis  of  x  when  x  is  positive,  and  convex  when  x  is 
negative. 

The  curve  has  neither  multiple  points,  cusps,  conjugate  points,  nor 
inflexions. 


2,  The  curve  whose  equation  is  if 


X*  —  a%2 


2x  —  a 

When  a:  =  0,  y  =  0,  and  therefore  the  curve  passes  through  the 
origin. 

When  X  =  -^  y  =zdt  ao  ,  when  x  z= -\-  oo ,  y  —  -{-  ao ,  and  when 
ar  =  —  oo,  y  =  —  CD  . 

Thus  the  curve  has  four  infinite  brancnes. 

When  X  =  a,  or  ar  =  —  a,  y  =  0  corresponding  to  two  interseo 
tions  with  the  axis  of  x. 


Since  ^  =  ^  renders  y  =  rt  oo  ,  there  is  one  asymptote  whose 

a 
equation  is  *  =  o* 


212  DIFFERENTIAL  CALCULUS. 

.   Also,  by  resclving  with  respect  to  y,  and  expanding,  we  get 


1  n 

=  -Y  (a?  +  -  -h  terms  involving  negative  powers  of  ^) 
y  =  —  [a:  +  -  J  is  the  equation  of  a  second  asymptote. 


Forming  the  value  of  the  differential  coefficient  — ,  we  have 


dy 
dy       6ar*  —  2a? x^  —  Aax^  +  2a  ^a: 

This  expression  becomes  infinite  when  x  =:  -^  when  a:  =  ±  a,  and 
when  a;  =  0. 

Hence  the  curve  cuts  the  axis  of  x  perpendicularly  at  the  origin, 

and  at  distances  therefrom  =  -f  a  and  —  a  respectively.     The  value 

dv 
of  v^  becomes   zero  when  6a;*  —  Aax"^  —  Sa^^e^  _j_  2a3^  __  q    which 
dx 

corresponds  to  a  value  of  x  between  0  and  —  a.     The  corresponding 

value  of  y  is  a  maximum. 

There  are  inflexions  at  the  points  where  a:  =  a  and  ic  =  —  a,  as  will 

readily  appear  by  substituting  for  x  values  alternately  a  little  greater 

and  somewhat  less  than  a,  and  similarly  for  values  greater  and  less 

than  —  a.     For   if  x   be   rather   greater   than   a   in   the   equation 

x^  —  (jfix^ 

y3  =  -— —iV  will  be  positive  ;  but  if  a;  be  somewhat  less  than 

<*a?  "~~  (I 

a,  y  will  become  negative.     Thus  the  curve  will  cross  the  tangent  at 


TRACING  OF   CURVES.  213 

the  point  where  it  meets  the  axis.  The  same  will  be  true  when  rr=  —a. 
There  will  be  a  third  inflexion  between  x  =  0  and  x  =z  -a,  for  the 
curve  touches  the  axis  of  y  at  the  origin,  and  a  parallel  asymptote  at 
the  distance  -  a  from  that  axis,  and,  therefore,  must  chan£;e  the 
direction  of  its  curvature  between  those  two  parallels. 

Finallv  by  making  the  value  of  -—  =  -we  shall  find  that  there 

(tec       \J 

is  a  cusp  of  the  first  kind  at  the  origin.     The  form  of  the  curve  is 
represented  in  the  diagram. 


PART  III. 

THEORY  OF  CURVED  SURFACES. 


CHAPTER    I. 

TANGENT    AND    NORMAL    PLANES    AND    LINES. 

173.  The  consideration  of  surfaces  affords  an  application  of  the 
vheory  of  functions  of  two  independent  variables.  Thus  if  x,  y, 
ind  z,  be  the  co-ordinates  of  any  point  in  the  surface,  and  z  =z  (p(x,y) 
*he  equation  of  the  surface,  the  values  of  x  and  y  may  be  assumed 
arbitrarily,  and  that  of  z  will  become  determinate. 

174.  Prop.  To  determine  the  general  differential  equation  of  a 
plane  drawn  tangent  to  any  curved  surface  at  a  given  point  {x^,  y^,  Zj) 
situated  in  the  surface. 

Let  the  surface  and  plane  be  intersected  by  planes  respectively 
parallel  to  xz  and  yz,  and  passing  through  the  point  (x^,  y^,  z-^). 

The  equations  of  the  line  cut  from  the  tangent  plane  by  the  plane 
parallel  to  ccz  will  be  of  the  forms 

x-x^  =  t{z-z^) (1),     and     y  =  y^ (2), 

and  those  of  the  intersection  parallel  to  yz  will  be  of  the  forms 

y  —  y,  =  s(g  —  ^i)  .  .  .  .  (3)     and     x  =  x^ (4). 

Also  the  equation  of  the  tangent  plane,  which  contains  these  lines, 
will  have  the  form 

A(x  -  X,)  -f  B(y  -yi)+C(z-z,)=0.,..  (5). 
The  equation  of  its  trace  on  xz  is  A(x—Xi)  =  —  C(z—z^)-\-jByi . .  (6). 
"     yz^'B(y-y,)=-C{z-z,)  +  Ax,.,(7), 


TANGENT  PLANES  TO  CURVED  SURFACES.      215 

But  the  trace  (6)  is  parallel  to  the  intersection  (1)  (2),  and  the  trace 

(7)  is  parallel  to  the  intersection  (3)  (4). 

C          ,  C 

.-.  t  = and      s  =  — ^« 

A  Jo 

which  values  reduce  (5)  to  the  form 

z-z^=z-(x-x^)-\--(i/  -y^) (8). 

Now  since  the  intersections  (1)  (2)  and  (3)  (4)  are  respectively 

tangent  to  the  corresponding  curves  cut  from  the  surface,  we  must 

,  dx,  ,  dy,  \       dz,  ,1       dz. 

have         t  —  ~~    and     s  =  -r-^     or    -  =  -—^     and      -  =z  -— ^» 
a^i  dz^  i      dXi  s      dy^ 

Hence  (8)  reduces  to 

z  —  z^  =  ■—-  (x  —  x^)  4-  — ^  {y  ~  l/i)  •  '  '  (^)»  t^G  desired  equation. 

The  expressions  -~  and  —^   are  the  partial  differential  coeffi- 
dx^  dy^ 

cients  derived  from  the  equation  of  the  surface,  and  they  will  have 
the  same  values  at  the  point  {x^^y^z^^  as  the  similar  coefficients  de- 
rived from  the  equation  of  the  plane,  tangent  at  that  point. 

175.    Cor.  If  the  equation  of  the  surface  be  given  under  the  form 
u  =  cp(x,  y,  z,)  =  0, 
the  equation  of  the  tangent  plane  will  take  a  more  symmetrical 
form.     For  we  then  have     (Art.  57) 

r^^^"}  _du      du   dz  rdu~\       du       du    dz 

Ldxj       dx      dz    dx  ~    ^  L  dyj  ~  dy       dz     dy  ~    ' 

du  du 

XT  dzy  dx-,      dz-,  dy-. 

Hence  — i  = -i,     -— ^  = fi, 

dx^  du       dy-^^  du 

dz^  dz^ 

and  by  substitution  in  (9)  and  reduction,  we  obtain  the  more  sym 
metri(;al  form 

/  \  ^^    ,    /  \  ^^    ,    /  \  du 


216  DIFFERENTIAL    CALCULUS. 

176.  Prop.  To  determine  the  equations  of  a  line  normal  to  a 
curved  surface  at  a  given  point  {x-^.y^z-^) 

The  equations  of  a  line  passing  through  the  point  {x^.y^^.z.).,  have 
the  forms 

x-x^  =  t{z-  0i),         y  -y^  =  s{z  -  z^)  ; 

ahd  since  the  normal  line  is  perpendicular  to  the  tangent  plane,  we 
have  by  the  conditions  of  perpendicularity  of  a  line  and  plane 
(-4  =  Ct  and  B  =  Cs),  the  following  relations: 

du  du 

.  _^  _        dz^  _  dx^  B             dz-^        dyj 

U            dx^        da  ~  C  ~        dy-^  ~  au 

dz^  dz-^ 

These  conditions  give  for  the  equations  of  the  normal  line 

L     or      ^      ^ 

,    dz   ,  \       r.\  \  du  .  ■       du  ,  . 

177.  Cor.  If  ^j,  ^2,  ^3,  he  the  angles  formed  by  the  normal  witl 
the  axes  of  .r,  y,  and  2,  respectively,  or  those  lormed  by  the  tangent 
plane  with  the  planes  of  yz.,  xz,  and  xy,  w^e  shall  have 

dz-^  du 

A  dx^  d.i\ 

*~.^/3M^i^2^6'2~        idzl^^dz^     ~~    /d^'~a^'~T^' 
V  d.i^^  "^  dy^^  "^        V  dx^^'^dy^^'^dz^^ 

dz^  du 

dyi  dy^ 


cosfl,  = 


Vdz^       dz-^  Idu^         du^        du^ 

^"^^■^  ydJ^^d^^^Jz'} 


cos  ^3  = 


du 
1  ~dz^ 


fdk^     dz.^        ,         [d^   , 


du'^       du^ 
d^'^d^^ 


TANGENT   PLANES  TO   CURVE  SURFACES.  217 

178.  Prop.  To  determine  the  equations  of  a  line  drawn  tangent 
to  a  curve  of  double  curvature,  at  a  given  point  (^i,yi,^i),  on  the 
curve. 

The  curve  will  be  given  by  the  equations  of  its  projections  on  two 
of  the  co-ordinate  planes,  as  arz,  and  yz\  thus 

F{x, ^)  ^  0,  .  .  .  .  (1).         and        9(y, ^)  =  0 (2). 

The  equations  of  the  required  tangent  will  have  the  forms 

X  -x^  =  t{z-z^), (8).     and     y  -  yi  =  s{z  -  z^), (4); 

and  since  the  projections  of  the  tangent  are  tangent  to  the  projections 
of  the  curve,  (-3)  and  (4)  will  take  the  forms 

^-^l  =  -^^{^-^l)^"{^)'      and      2_^j  =  -^-l(y-yj),  ..  (6). 

in  which  equations  the  values  of  -r-^  and  -—  are  to  be  derived 

dx-^  dy^ 

from  (1)  and  (2),  the  equations  of  the  given  curve. 

179.  Prop.  To  determine  the  equation  of  a  plane  drawn  through 
a  given  point  of  a  curve  of  double  curvature,  and  normal  to  the 
curve  at  that  point. 

The  equation  of  a  plane  passing  through  the  point  (^1,^1,2^1),  is  of 

the  form 

A{x-x,)  +  B{y  -  y,)  +  C{z  -  z,)  =  0.  .  ,  .  (1). 

But,  since  the  plane  is  to  be  perpendicular  to  the  tangent  line,  we 
must  have  the  conditions 

^  =  (7/=C^,     and     ^=:(7s=e^\ 
dzy;  dz^ 

which  values  reduce  (1)  to  the  form 

(-— .)|;+(2/-.,)|+(^-..)=o,     . 

the  required  equation. 


218  DIFFERENTIAL   CALCULUS. 

EXAMPLES    OF    TANGENT    PLANES    TO    SURFACES. 

180.  1.  The   tangent   plane   to   the    sphere   whose   equation    ia 

«*  =  iC^  -f-  2/2  _|_  ^,2  —  ^2  __  Q^ 

XT  ^^       ^        da       ^        du       ^ 

Here  -—  =z  2a;,     -r-  =  2y,     -r-  =  ^z. 

ax  ay  dz 

Therefore  by  substitution  in  the  general  differential  equation  of  a 
tangent  plane  to  a  curved  surface,  we  get 

-\-2z^{z-z^)=0. 
.  • .    xx^  4-  yVi  +  ^^i  =  ^\  4-  2/i^  +  z-^  =  r^,  the  required  equation. 

2.  The   ellipsoid      ifrir^  +  C  +  ^-l^O. 
a^        b^        c^ 

du  _  2x      du       2y      du       2z 
^""^'     dij"^!^'     ~dz^'^' 

•  •  •  ^(^^  -  ^0  +  T2'(y  -  yi)  +  ^IH^  -  ^i)  =  0. 


^2W  71/      .       ^2 


/»•'>•         ?y?y         zz 
or,      — --  +  ^  -I — ^7  =  1,  the   required   equation  of   the   tangent 
a^         b^        c^ 

plane. 

x'^ 

rhp.  livnorlinlnirl  nf  r»nA  sViPP.t   ->/  =r  —  -I- 

b'^         C2 


iK"^  w^         2f2 

3.  The  hyperboloid  of  one  sheet  ^  =  -t+75 5 1  =0. 


du      2a;      du      2y      c??/  2^ 

^  ~  '^'     Ty~l^'     1z^~"^' 

.  • .   -y  +  ^|- ^*  —  1  =  0,  the  equation  of  the  tangent  plane. 


CYLINDRICAL  SURFACES.  219 

4.  The  conoid     u  =  c^-x^  +  y  V  _  ^2^2  _  0. 

<^u       ,  „        du       ^  „        du       _  „        ^  ^ 
dx  '     rfy  *"      c/2         ^ 

or,     c^ar^i  +  z-^yyy^  +  (y^^  _  ^2)^2^  =  y^z^^    the   equation   of   the 
tangent  plane. 


CHAPTER    II. 


CYLINDRICAL    SURFACES,    CONICAL    SURFACES.    AND    SURFACES    OF 
REVOLUTION. 

181.  Prop.  To  determine  the  general  differential  equation  of  all 
cylindrical  surfaces. 

These  surfaces  are  generated  by  the  motion  of  a  straight  line, 
which  touches  a  fixed  curve,  and  remains  parallel  to  a  fixed  line 
in  every  position. 

Let  the  equations  of  the  fixed  curve  or  directrix  be 

F{x,z)=.0,,...  (1).         i^,(y,.)=0,....(2), 
those  of  the  generatrix,  in  one  of  its  |)ositions,  being 

x  =  tz  +  a, (3).         y=:sz  +  b, (4). 

Since  the  generatrix  continues  parallel  to  a  fixed  line,  the  values 
of  t  and  5  will  continue  constant  for  all  positions  of  the  generatrix, 
but  a  and  b  will  vary  with  its  position. 

Eliminating  x  between  (1)  and  (3),  and  y  between  (2)  and  (4), 
we  get  one  rehition  between  z  and  cr,  and  a  second  between  z  and  6. 
Then  combining  these  equations  to  eliminate  2,  we  obtain  a  relation 
between  a  and  /;,  which  may  be  written 

6  =  9c/,  ...  .  (5). 


220  DIFFEliENTIAL   CALCULUS. 

But  from  (3)  and  (4),     a  z=l  x  —  tz,     and     b  =  i/  —  sz. 

,' .  (5)  becomes    y  —  az  =  ip(x  —  tz),  ....  (G). 

This  is  a  general  equation  of  all  c\lindrical  surfaces,  but  it  con- 
tains the  unknown  function  cp.  To  eliminate  this  function,  differen- 
tiate (t))  with  respect  to  x  and  y  successively,  and  divide  the  first 
result  by  the  second  ;  thus 

dz        d:(x  —  tz)       d(x  —  tz) 
dx  ~    a{x  —  fz)  dx 


and 


dz       d  (x  —  tz)       d{x  —  tz) 
dy        d[x  —  tz)  dy 


dz  dz 

dx  dx 


<iz  dz 

1  —  if  —  —  t  — 

dy  dy 

whence       i— -  4-  ^-y-  =  1  .  •  •  .  ("7),     the  required  equation. 

182.  Cor.  if  we  denote  the  primitive  or  integrated  equation  of  a 
cylindrical  surface  hy  f{x^y^z)  z=z  u  =i  0  the  differential  equation  (7) 
may  be  reduced  to  a  more  symmetrical  form.     For  since 

du  dn 

dz            dx  J      ^^  _       ^y 

dx            da  dy             du 

dz  dz 

we  obtain  by  substitution  in  (7)  and  reductior. 
dv  da       du        .  ,^. 

a  form  often  more  convenient  than  (7). 

183.  Pr<jp»  To  determine  the  equation  of  the  cylindrical  surface 
which  envelops  a  given  surface,  and  whose  axis»  is  parallel  to  a 
given  line. 

The  enveloping  and   enveloped  surfaces  being  tangent  to   each 


CYLINDRICAL  SURFACES.  221 

Other,  will  hiave  a  common  tangent  plane  at  every  point  in  the  curve 
of  contact,  and  the  equation  of  one  of  these  planes  will  be 

in  which  z^  y^  z^  refer  to  a  point  of  contact.     Moreover  the  differen- 

,  ,         a-  '  dzi   dz-i  du     du     du  .  ,     ,       , 

tial  coemcients   —-^5  ——  or  --. — ?  - — »  - —  are  the  same  whether  de- 
axy   ayj         ax^    dy-^    dz^ 

rived  from  the  equation  of  the  cylinder  or  from  that  of  the  enveloped 
surface.  Hence,  if  we  form  the  differential  coefficients  from  the 
equation  of  the  given  surface,  and  substitute  their  values  in  the  dif- 
ferential equation  of  the  cylinder,  the  result  will  characterize  the 
points  of  contact,  being  the  equation  of  a  surface  containing  those 
poilits.  This  equation,  when  combined  with  that  of  the  enveloped 
surface,  will  give  the  equations  of  the  curve  of  contact,  and  ihence 
the  cylinder  can  be  determined. 

184.  ^Jc.  A  sphere  M  =  x^  -{-  y^  -^  z^  —  r^  =0  is  enveloped  by 
a  cylinder  whose  axis  is  parallel  to  the  axis  of  ^ ;  to  find  the  curve 
of  contact. 

Here  we  have  ar  =  a  the  equation  of  the  projection  of  the  generatrix 
on  xzj  and  y  =  6  the  equation  of  the  projection  of  the  generatrix  on  yz, 
.'.  t  =  0,     s  =  0. 

Ai  du       ^        dti       ^        du       ^ 

^'^°  ^  =  2^'    Ty  =  ^^'   ^  =  ^- 

Hence  by  substitution  in  (8), 

0.2x  -\-0.2y  +  2z  =  0    or    «  =  0, 

and  the  points  of  contact  all  lie  in  the  plane  of  xy. 

Combining   the    equations    x'^ -{- y^ -\- z"^  —  r^  =z  0    and    5  =  0, 

there  results 

a;2  -(-  y2  __j.2  _  0. 


222  DIFFERENTIAL  CALCULUS. 

.  • .  The  curve  of  contact  is  a  great  circle  of  the  sphere,  as  might 
have  been  foreseen. 

185.  P/o/j.  If  any  surface  of  the  second  order  be  enveloped  by  a 
cylinder,  the  curve  of  contact  will  be  an  ellipse,  hyperbola  or  para- 
bola, or  a  variety  of  one  of  those  curves. 

The  general  equation  of  surfaces  of  the  second  order  is 

.•.^  =  Dz-i-2Ex-\'Fy-^l     p  =  Bz-^2Cy  +  Fx  +  H, 

du 

~=  By  +  2Az-\-  I)x-{-  G. 

'   '  ^Tx  "^'^  ^  Tz  -<^^+2J5'^+i^y+/)  ^s{Bz^2Cy-^Fx^H) 
+  (%  +  2.42H-i>ar+  (?)  =0, 

which  is  the  equation  of  a  plane. 

Hence  the  points  of  contact  are  confined  to  one  plane.  But  any 
section,  by  a  plane,  of  the  surface  represented  by  the  equation  (1). 
will  necessarily  be  a  line  of  the  second  order,  and  therefore  the 
truth  of  the  proposition  is  apparent. 

Conical  Surfaces. 

186.  Prop.  To  determine  the  general  differential  equation  of  all 
conical  surfaces. 

These  surfaces  are  generated  by  the  motion  of  a  straight  line 
which  touches  constantly  a  fixed  curve  snd  passes  through  a 
fixed  point. 

Let  the  equations  of  the  directrix  be 

i^{:r,^)  =  0  .  .  .  .  (1),     F,(,j,z)=0....(i); 

those  of  the  generatrix  in  one  of  its  positions  being 

t 
X  —  a  —  t{z  —  c)  .  .  ,  .  (3),     and     y  —  b  =  s{z  —  c)  .  ,  ,  .  (4). 

where  a,  b,  and  c,  denote  the  co-ordinates  of  the  fixed  point  or  vertex. 


CONICAL  SUKFACES.  223 

The  quantities  t  and  s  vary  with  the  position  of  the  generatrix, 
hut  a,  6,  and  c,  are  constant. 

Eliminating  x  between  (1)  and  (3),  and  y  between  (2)  and  (4),  we 
i:et  one  relation  between  z  and  /,  and  a  second  between  z  and  s. 
Then  combining  these  equations  to  eliminate  0,  we  obtain  a  relation 
between  t  and  s,  which  may  be  written 

s  —  (^t (5). 

But  from  (3)  and  (4),  t^  ^  ~  ^,     and     s  =  ^^^^. 
^  '  z  —  c  z  —  c 

.  • .   (5)  becomes         — =  9    (6). 

This  is  an  equation  of  conical  surfaces,  but  it  contains  the  unknown 
function  (p.     To  eliminate  this  function,  differentiate  (6)  with  respect 
to  X  and  y  successively,  and  divide  the  first  result  by  the  second ; 
thus 
_  y  —  fe     dz d^[^  ]        d[  ]  _  dcp[  ]      pi  X  —  a      dz 


[1  X  —  a      dz\ 

z—G      {z  —  cY    dxj 


{z-cf   dx         d[]  dx         d^]      Lz-G  {z  —  cf 

and 

1  y  —  h      dz  __  G?p[  ]       c?  [  ]  _  c?(p[  ]        r       x  —  a    dzl 

TTc  "■  (z  -  cf  '  Jy  ~  Tf  ]  ^  ~di'~  ~d[]  ^  L  (z-cfd^T 

1. 

Now  by  division 


.  .    dz  .  dz 

dz  dz 

.  • .  2  —  c  =  (a;  —  a)  —  4-  (y  —  b)  —  •  •  •  (7)  the  required  equation. 

187.  Cor.  If  we  denote  the  primitive  or  integrated  equation  of  a 
conical  surface  by  /  {x,  y,  z)  z=  u  =:  0,  the  differential  equation  (7) 
may  be  reduced  to  a  more  symmetrical  form. 


du 

du 

dx 

and 

dz 

dy 

du' 

dy- 

'  du' 

dz 

dz 

224  DIFFERENTIAL  CALCULUS. 


■n.        .  dz 

Jjor  since      —r- 

dx 


we  obtain  by  substitution  in  (7)  and  reduction 

{x-a)-~^^-(y-b)-~-\-{z--c)-^^=:0 (8),  a  form  often 

more  convenient  than  (7). 

188.  Prop.  To  determine  the  equation  of  the  conical  surface  which 
envelopes  a  given  surface,  and  whose  vertex  is  situated  at  a  given 

point. 

,      ,.^         .1        r^  .  dz         ^  dz       du  du       ^  du 

If  we  form  the  differential  coefficients  -7-,  and  — —  or  -p,  -7-  and  -7- 

dx  dy       dx  dy         dz 

from  the  equation  of  the  given  surface,  and  substitute  their  values  in 

(7)  or  (8),  the  differential  equation  of  the  conical  surface,  the  resulting 

relation  will  characterize  the  points  of  contact,  being  the  equation  of 

a  surface  which  contains  those  points.     This  equation,  combined  with 

that  of  the  enveloped  surface,  will  give  the  equations  of  tiie  curve 

of  contact,  and  thence  the  cone  can  be  deterrnined. 

Ex.  A  sphere  a;^  -f  y2  _j_  2-2  _  ^2  _  q  _.  ^^  is  enveloped  by  a  cone 
whose  vertex  is  situated  on  the  axis  of  y,  at  a  distance  h  from  the 
origin;  to  find  the  curve  of  contact. 

Here  we  have  the  co-ordinates  of  the  vertex  a  =  0,  6  =  6,  c  =:  0, 

Also,  — -  =  2ar,  — -  =  2y,  -7-  =  22. 

'  dx  dy  dz 

.  •.  By  substitution  in  the  equation  of  conical  surfaces 

{x  _  0)  2^  +  (y  -  ^)  2y  +  (0  -  0)  2^  =  0  ; 

or,  ic2  -f  y2  ^  2^  _  5y  _.  0. 

This  being  the  equation  of  a  sphere  having  a  radius  =  -^,  and  its 


CONICAL  SURFACES.  225 

centre  on  the  axis  of  y  at  a  distance  -b  from  the  origin,  the  points  of 

contact  must  lie  in  the  surface  of  such  a  sphere. 

By  combining  the  equations  of  the  two  spheres,  we  get 

by  =  7"^  or  y  =z  — —  and  x^ -\-  z^  —  — —  (62  —  r^). 
6  O'^ 

Hence  the  curve  of  contact  is  a  circle  perpendicular  to  the  axis  o* 

y,  and  at  a  distance  -7-  from  the  origin. 

189.  Prop.  If  any  surface  of  the  second  order  be  enveloped  by  a 
cone,  the  curve  of  contact  will  be  an  ellipse,  hyperbola,  or  parabola, 
or  a  variety  of  one  of  these  curves. 

The  general  equation  of  surfaces  of  the  second  order  is 

Az'^-\-Bzy+Cy'^  +  Dzx^Ex'^-[-Fxy+Gz+Hy-[-Ix+K=:0=u..{\). 

(Lit  UlJ 

.'.—=Dz+Fy-\-^Ex+I,     —^Bz^Fx-^'^Cy^-H, 
^  =  By -{- Ifx  +  2Az -{-  G. 

=  [Bz  -hFy+  2Fx+I]  {x-a)-i-  [Bz  +  Fx -{-  2Cy  +  H]  (y-b) 
-f  [By  -{-  Bx  -{-  2Az  -\-  G]  {z  —  c)  =  0, 

or,  2[Az^  +  (7y2  -^  Ex^  +  ^[Bzy  +  Bzx  +  Fxy] 

+  [G  -  Ba-  Bb-  2Ac]z  +  [H  -  Fa  -  Be  -  2Cb]y 
+  [I-Fb  -Be-  2Fa]x-  [Gc  -{- lib -^  la]  =0  . . .  (2). 
By  combining  (1)  and  (2),  we  get 

[G  +  Ba -\-  Bb -\-  2Ac]  z  +  [H  +  Fa -\-  Be -\-  2Cb]y 
-f  [I  -\-  Fb  +  Be  +  2Fa]x  -h  2K  -\-  Gc -^  lib -{- la  =  0. 
This  is  the  equation  of  a  plane,  and  therefore  the  curve  of  contact 
is  the  intersection  of  the  given  surface  by  a  plane,  and  consequently 
an  ellipse,  hyperbola,  or  parabola. 
15 


226  DIFFERENTIAL   CALCULUS. 

190.  Prop.  To  determine  the  general  differential  equation  of  all 
surfaces  of  revolution. 

Let  a:  =  ^2  +  a  .  •  .  .  (1)  )  ,     .,  ,.  ^  ^u 

^  '^   J.  be  the  equations  of  the  axis. 

y=^sz-\-h (2)  j 

J^(:r,0)  =  0  .  .  .  .  (3),     and     F^{y,z)  =  ^  .  .  .  .  {\\ 
those  of  the  generatrix. 

The  characteri^ic  property  of  this  surface  is,  that  every  plane 
section  perpendicular  to  the  axis  is  a  circle.  Now  the  equation  of  a 
plane  perpendicular  to  the  line  (1)  (2)  is 

z-\-  tx  ^  8y  —  c, 
and  the  circle  cut  from  the  surface  by  this  plane  may  be  supposed 
situated  on  the  surface  of  a  sphere  whose  centre  may  be  assumed  at 
any  point  on  the  axis,  and  whose  radius  will  be  determined  by  the 
value  of  c,  when  the  centre  has. been  chosen. 

Take  the  centre  of  the  sphere  at  the  point  (a,  i,  0),  where  the  axis 
pierces  the  plane  of  xy^  and  the  equation  of  the  sphere  will  be 

{x  -  ay  4-  (y  —  by  +  22  _  j.2^ 
But  r  and  c  are  mutually  dependent  upon  each  other,  which  fact 
may  be  indicated  by  the  equation  c  =  (p(r^).     Hence 

z  +  tx-i-sy  =  (p[{x-  af  Ar  {y  -  bf -\- z'^'] (5). 

To  eliminate  the  unknown  function  9,  differentiate  (5)  with  respect 
to  y  and  x  successively,  and  divide  the  first  result  by  the  second. 

.    ^4.,_MJx^    and     ±  +  ,-<^*n^''[]. 
•    ■'  J      TO  —    jr  T    A  -3         ana      — — \-  i  —  x  ^— r 

ay  dy  \  ay  dx  dy  \         ax 

dz  r    :      ^^ 

.    dy __''_ dy_^ 

'    '  dz  dz' 

dx  dx 

or       {x-a-tz)~-(y-b-sz)-£  ^{x-a)s-{y^b)t=0  .  .  .  (6), 
which  is  the  required  equation  of  surfaces  of  revolution. 


SURFACES   OF   REVOLUTION. 


227 


Cor.  When  the  axis  of  revolution  coincides  with  that  of  z^  we  have 

^  =  0     and     .<?  =  0,     a  =  0     and     6  =  0. 

dz  dz 

.  •.  (6)  reduces  to        a:— -y—  =  0....  (7). 

191,  Prop.  A  given  curved  surface  revolves  about  a  fixed  axis ; 
to  determine  the  surface  which  touches  and  envelopes  the  moveable 
surface  in  every  position. 

The  required  surface  will  obviously  be  a  surface  of  revolution, 
whose  generatrix  will  be  the  curve  of  contact  of  that  surface  with 
one  of  the  moveable  surfaces. 

Hence  if  we  determine  the  values  of  the  differential  coefficients 

-—  and  -r-  from  the  given  surface,  and  substitute  them  in  the  gene- 
ix  dy  ° 

ral  differential  equation  of  all  surfaces  of  revolution,  the  result  will 
characterize  the  points  of  contact,  being  the  equation  of  a  surface 
containing  those  points.  This  equation,  combined  with  that  of  the 
given  surface,  will  give  the  equations  of  the  curve  of  contact  or  the 
required  generatrix. 

192,  1.  A  right  cone  with  a  circu- 
lar base,  whose  vertex  is  at  the  origin, 
and  whose  axis  coincides  originally 
with  the  axis  of  x^  is  caused  to  re- 
volve about  the  axis  of  ^ :  to  deter- 
mine the  form  of  the  enveloping  sur- 
face. 

Put  the  semi-angle  ^OC  of  the 
:one  =  v,  and  tan  v  —  t. 

Then  the  equation  of  the  cone,  in 
the  position  AOB  will  be 


^2     _^     y2     ^     fly.2^ 

dz 


—     and 


s2  =  t^x^ 
dy 


(1) 


228  DIFFERENTIAL   CALCULUS. 

which  values  substituted  in  the  differential  equation  of  surfaces  of 

revolution,  viz. 

dz  dz        ^        .  xy   ,    a^xy 

x—-y—  =  (i^    gives    -^H ^  =  0. 

dy         dx  z  z 

,' .  x  =1  0     or     y  =  0. 

Combining  the  first  of  these  results,  a;  =  0,  with  the  equation  of  tlie 
cone,  we  get 

^2  _|_  ^2  _  Q^       .    2;  =r  0     and     y  =  0, 

which  conditions  apply  to  the  origin  exclusively  ;  but  the  second 
result  y  =  0,  gives  by  combination  with  (1) 

z"^  r=  fx"^'     or     z  z=  diz  tx     and     y  =  0, 

which  are  the  equations  of  the  lines  OA  and  OB. 

Hence  the  required  envelope  is  a  double  cone  generated  by  the 
revolution  of  the  lines  OA  and  OB  about  OZ. 

2.  A  sphere  (.r  —  a)^  -f-  (y  —  i)2  -|-  ^^  —  r^,  revolves  about  the 
•xis  of  2  ;  to  find  the  enveloping  surface.     Here  we  have 


dz 

X  —  a 

,      dz            y  —  h 

and      J-  = 

dy                 z 

dx~ 

z 

dz 

dz 
-^d.= 

xy  —  hx    1    xy  —  ay 

z                    z 

-=0. 


.  • .  bx  —  ay  =  0^  the  equation  of  a  plane  passing  through  the 

axis  of  2;,  and  the  centre  (a,6)  of  the  sphere. 

This  plane  intersects  the  sphere  in  a  great  circle,  whose  equation, 

in  its  own  plane,  is 

{r,-a,Y  +  z'^  =  r\ 

Ji  which  r^  =  x"^  +  y"^,     and     a^  =  «2  _j_  52^ 

.  •  .    [(a;2  -f  y2)^_  (^2  ^  52)^]%  ^2  ^  r\ 

or,         a;2  +  ^2  +  .2  _  ^(cfl  +  62)*(.'r2  +  y^f  =:  r^  ^  d^  —  h\ 
the  equation  of  the  required  surface. 


SURFACES  OF  REVOLUTION.  229 

When  o?  -\-  h'^  =  r^,  this  reduces  to 

^2  _^  y2  _j_  ^2  _  2r(a:2  _|.  2/2)*=  0  ; 
and  when  a  =  0,  b  =  0,  x^  +  y"^  -{-  z"^  =  r"^^ 
the  equation  of  the  sphere. 

/|j2  y'2i  ^2 

3.  An  ellipsoid  — -f^ -}-—=:  1,  revolves  about  the  axis  of  y ; 

^  a^      0^       c^ 

to  determine  the  enveloping  surface. 

The  differential  equation  of  the  surface  is,  in  this  cas 


Also, 


dx         dz 

dy 
dx~ 

b^x       dy            b^z 
d^y^      dz  ~       c^y 

hH         b'^z      ^ 

Z-T-  +  X—   z=:  0,       .'.    XZ  =  0, 

ay         cy 


and  consequently         x  =  0,         or,         z  =  0. 

y^       z^ 
But  when  x  =  0,  —-{ — j^l?  an  ellipse  in  the  plane  of  yz, 

x^    '    y"^ 
And  when  2  =  0,  ~+— =1,  an  ellipse  in  the  plane  of  xy. 


Hence  the  required  envelope  consists  of  two  ellipsoids  of  levolu. 
lution,  whose  equations  are 


CHAPTER  III. 

CONSECUTIVE  SURFACES  AND  ENVELOPES. 

193.  Ill  the  last  chapter  we  have  presented  some  examples  of 
surfaces  enveloping  a  series  of  other  surfaces,  but  in  the  only  case 
considered,  the  enveloped  surface  was  supposed  to  be  of  invariable 
form,  and  its  change  of  position  was  effected  only  by  a  revolution 
around  a  fixed  axis.  In  that  case,  the  enveloping  surface  was  neces- 
sarily a  surface  of  revolution. 

It  is  now  proposed  to  consider  the  envelopes  to  any  series  of  con- 
secutive surfaces. 

194.  If  different  values  be  successively  assigned  to  the  constants 
or  parameters  which  enter  in  the  equation  of  any  surface,  the  several 
relations  thus  produced,  will  represent  as  many  distinct  surflices, 
differing  from  each  other  in  form,  or  in  position,  or  in  both  these 
particulars,  but  all  belonging  to  the  same  class  or  family  of  surfaces. 
When  the  parameters  are  supposed  to  vary  by  infinitely  small  in- 
crements, the  surfices  are  said  to  be  consecutive. 

Thus  let  F{x^  y,  2;,  a)  =  0,  ....  (1),  be  the  equation  of  a  surface, 
and  let  the  parameter  a,  take  an  increment  A,  converting  (1),  into 
F(x,  y,  z,  a  -{-  h)  =  0,  .  .  .  .  (2) ;  then  if  h  be  supposed  indefinitely 
small,  the  surfaces  (1)  and  (2)  will  be  consecutive.  Moreover,  the 
surfaces  (1)  and  (2)  will  usually  intersect,  and  their  intersection  will 
vary  with  the  value  of  h,  becoming  fixed  and  determinate  when  the 
surfaces  are  consecutive.  * 


CONSECUTIVE   SURFACES  AND   ENVELOPES.  281 

195.  Prop.  To  determine  the  equations  of  the  intersection  of  con- 
secutive surfaces. 

To  effect  this  object,  we  must  combine  the  equations 

F{x,y,z,a)  =  0,  .  .  .  ,  (1);      and     F{x,y,z,a-^  A)  =  0,  .  .  .  .  (2), 

and  then  make  h  =  Q. 

By  reasoning  precisely  as  in  the  case  of  consecutive  curves, 
(Art   143)    we   prove  that  the  two  conditions 

i^(^,y,.,a)  =  0,....(l),        and        ^^^iMI  =  0,  .  .  .  .  (3), 

must  be  satisfied  at  the  same  time. 

By  combining  these  equations,  so  as  to  eliminate  first  y,  and  thes 
a:,  we  shall  have  the  equations  of  the  projections  of  the  required  in- 
tersection on  xz^  and  yz. 

196.  Prop.  The  surface  which  is  the  locus  of  all  the  intersections 
of  a  series  of  consecutive  surfaces,  touches  each  surface  in  the 
series. 

If  we  eliminate  the  parameter  a  between  the  two  equations 

F{x,y,z,a)  =  0 (1),     and     5^'-^)  =  0,  .  .  .  .  (2), 

the  resulting  equation  will  be  a  relation  between  the  general  co-ordi- 
nates x^  y,  2;,  of  the  points  of  the  various  intersections,  independent 
of  the  particular  curve  whose  parameter  is  a,  or  in  other  words,  the 
equation  of  the  locus. 

Resolving  (2)   with  respect  to  a,  the  result  may  be  written 

a  =  (p(x,  y,  z), 

and  this  substituted  in  (1)  gives 

F[x,y,z,cp{x,y,z)]  =  0,  .  .  .  .  (3), 

which  will  be  the  equation  of  the  locus. 

Now  differentiating  both  (1)  and  (3)  first  with  respect  to  x,  and 
then  with  respect  to  y,  we  readily  prove,  precisely  as  in  the  case  of 


232  DIFFERENTIAL   CALCULUS. 

dz  dz 

consecutive  curves,  that  the  values. of  -7-    and   -7-    are  the  same 

dx  dy 

whether  derived  from  (1)  or  (3).     Hence  the  two  surfaces  (1)  and 

(3)  will  have  a  common  tangent  plane,  and  will  therefore  be  niutu 

ally  tangent  to  each  other  at  all  points  common  to 'those  surfaces. 

197.  The  surface  (3),  which  touches  each  surface  of  the  series,  is 
called  the  envelope  of  the  series. 

198.  Ex.  To  determine  the  envelope  of  a  series  of  equal  spheres 
whose  centres  lie  in  the  same  straight  line. 

Assuming  the  line  of  centres  as  the  axis  of  ar,  the  equation  of  one 
of  these  spheres  will  be  of  the  form 

(^  _  a)2  _^  2^2  4.  ^2  _  ,.2  ^  0 (1), 

in  which  a  is  the  only  variable  parameter. 
Differentiating  with  respect  to  a  we  get 

—  2a:  +  2a  =  0 (2). 

From  (2)  a  =r  a:,  and  this  substituted   in  (1)  gives 

2/2  _|_  ^2  _  ^2   _  0^ 

This  is  the  equation  of  a  right  cylinder  with  a  circular  base,  the  axis 
of  which  coincides  with  that  oi  x.  •       ' 

199.  When  the  equation  of  the  proposed  surface  contains  two 
parameters  a,  6,  independent  of  each  other,  we  must  have  the  three 
conditions 

F(.,y,.,a,l,)  =  0  .  .  .  (1),      '-^^^^^^  =  0  .  .  .  .  (2). 

and  ^^Lfl^  =  0....(3). 

db  ^ 

And  by  eliminating  a  and  b  between  (1),  (2),  and  (3),  the  equation 
of  the  required  envelope  will  be  obtained.  Also,  if  the  proposed 
equation  should  contain  three  or  more  parameters  a,  b,c,  &c.,  two  of 
which,  a  and  b,  are  arbitrary,  and  the  others  connected  with  them 


CONSECUTIVE   SURFACES   AND   ENVELOPES.  233 

by  given  relations,  such  relations  will  enable  us  to  eliminate  the  ad- 
ditional parameters  and  to  obtain  a  final  equation  between  x,  y,  and  z. 

jC        V        z 

200.   1.  A  plane  whose  equation  is  — h  t  +  -  =  1?  Js  touched  in 
^.  a       0       c 

every  position  by  a  surface,  the  variable  parameters  a,  6,  and  t'being 

connected  by  the  relation  ahc  z=z  m^  :  to  determine  the  equation  of 

the  surface  or  envelope. 

From  -  +  r-l 1=0..  ..(1)  we  obtain  by  differentiation. 

a       0       c  ^  '' 

regarding  a  and  h  as  independent,  and  c  dependent  upon  them, 
X         z     dc         ^  ,^,  ,  y       z     dc       ^  ,^^ 

But  the  condition  ahc  =  m^  .  .  .  .  (4)  gives  by  dllferentiation 

dc  dc 

be  -{-  ab  -J-  =  0,     and     ac  -}-  ab  —-  =  0. 
da  ab 

dc  c  ^     dc  c 

.'.  —  = 5     and     -—=:—-. 

da  a  db  b 

which  values  substituted  in  (2)  and  (3)  reduce  them  to  the  forms 

X       z     c        ^  -,  y      z    c        . 

,  X      z  ^      y       z       . 

whence  -  =  -     and      -  =  — 

a       c  be 

These  Values  in  (1)  £?ive      -  H f  -  =  1, 

^  c        c       c 

or  —  =  1.     .  • .  c  =  32. 

c 

And  similarly  b  =  3y,     a  =  Sx. 

Finally  by  replacing  a,  b,  and  c,  in  (4),  by  their  values  just  found,  we 

obtain  xyz  =  —  as  the  equation  of  the  enveloping  surface. 

2.  To  find  the  envelope  of  all  the  spheres  whose  centres  lie  in  the 


^34  DIFFERENTIAL    CALCULUS. 

same  plane,  and  whose  radii  are  proportional  to  the  distances  of 
their  centres  from  a  fixed  point  in  that  plane. 

Assuming  the  plane  of  the  centres  as  that  of  xy^  and  the  origin  at 
the  fixed  point,  the  equation  of  one  of  the  spheres  will  take  the  form 

{x  -  af  +  (y  _  6)2  4-  s2  _  ,.2  ^  0 (1), 

in  which  a,  5,  and  r,  are  variable  parameters,  a  and  b  being  inde 
pendent,  and  r  connected  with  them  by  the  relation 

r2  =i:  t'^{a?  +  b"^)  .  .  .  .  (2)     where  t  is  a  constant. 

Eliminating  r  between  (1)  and  (2)  we  have 

(x  -  a)2  +  (3/  -  by  4-  s^  -  t\o,'^  4-  ^^)  =  0 (3). 

Differentiating  with  respect  to  a  and  b  successively, 

_  {x-  a)  -fa  =  (^  ..  .  (4),    and     -  {y  —  b)  -  t%  =  0  .  .  .  (5). 

X  It 

,'.  a  ■=! ^,     and     b  — ;     which  values  in  (3)  give 


(a;2  +  y2)(^2  _  ^4)  ^  2^2  (1  -  ^2)2     or     a;2  4-  y2  =.  — —  z^. 


1  -f 


This  is  the  equation  of  a  right  cone  with  a  circular  base,  its  axis 
being  coincident  with  that  of  2,  and  its  vertex  at  the  origin. 


CHAPTER   IV. 


CURVATURE    OF    SURFACES. 


201.  Two  surfaces  are  said  to  be  tangent  to  each  other  when  they 
have  a  common  point,  {x,  y,  z,)  and  a  common  tangent  plane  at  that 
point. 

Let  the  equations  of  the  two  surfaces  be 

F(X,Y,Z,)=0..,.(1),     and     (p(a:,y,^)  =  0  .  .  .  .  (2). 

The  analytical  conditions  necessary  for  a  simple  contact,  or  contact 
of  the  Jirst  order,  are 

„  ,_.  ^  dZ        dz      dZ         dz 

'  ^'  '  dX        dx\  dY        dy 

If  the  second  differential  coefficients,  derived  from  the  equations  of 
the  two  surfaces  be  also  equal,  viz.  : ' 

d'^Z       dH      d?Z        d'^z  ,        d'^Z  d'^z 

and 


dX^~dx^'     dY^~  dy'  dXdY        dxdy  ' 

the  contact  is  said  to  be  of  the  second  order.     If  the  third  differential 
coefficients  be  also  equal,  the  contact  is  of  the  third  order,  &c. 

202.  In  order  to  show  that  the  contact  will  be  more  intimate  as 
the  number  of  equal  differential  coefficients  becomes  greater,  let  the 
arbitrary  increments  h  and  k  be  given  to  the  independent  variables, 
X  =  a:  and  Y  =  y,  converting  Z  and  z  into  Zj  and  z^,  we  shall  then 
have    (Art.  82) 

'-    ^dX'l   'dY'\^dX^'l.2^  dXdY  '  I'^dY^'Trz'^' 


236  DIFFERENTIAL   CALCULUS. 

dz   h       dz   h       d'^z     K^  d^z     hh       d'^z     k^ 

and  when  Z  zzz  z. 

VdZ     dzlh     VdZ     dzlk     nPZ     d^zl    h^ 

^^ ~ ^' - \jlx~dx\\^ UY'TyAi'^ L'dX^'~d^^\Tyz  ^'  ^''' 

Now  the  value  of  this  difference  will  depend  (when  h  and  k  are 
very  small),  chiefly  on  the  terms  containing  the  lowest  powers  or 
h  and  k.  If,  therefore,  the  first  differential  coefficients,  derived  from 
the  equations  (/I),  (^),  and  (C),  of  three  surfaces,  at  a  common 
point,  be  equal,  and  if  the  second  differential  coeflicients,  derived  fiom 
(^4)  and  (^),  be  also  equal,  but  those  of  [A)  and  (6')  unequal,  the 
surfaces  {A)  and  [B)  will  separate  more  slowly,  in  departing  from 
the  common  point  than  will  the  surfa<-es  (^A)  and  (C). 

203.  The  order  of  closest  possible  contact  between  one  surface 
entirely  given,  and  another  given  only  in  species,  will  depend  on  the 
number  of  arbitrary  parameters  contained  in  the  equation  of  the 
second  surface. 

Thus  a  contact  of  the  first  order  requires  three  conditions,  and 
therefore  there  must  be  three 'arbitrjiry  parameters.  A  contact  ol 
the  second  order  requires  six  parameters;  one  of  the  third  order, 
ten  parameters,  &c.  Hence  the  plane,  whose  equation  has  three 
parameters,  may  have  contact  of  the  first  order.  The  sphere  cannot, 
except  at  particular  points,  have  contact  of  the  second  order,  since 
its  equation  has  but  four  parameters ;  but  of  two  tan^^'i'-  spheres, 
one  may  have  closer  contact  than  the  other. 

The  ellipsoid,  hyperboloid,  and  paraboloid,  can  eac)'  have  contact 
of  the  second  order. 

204.  Prop.  To  determine  the  radius  of  curvat«  re  of  a  normal 
section  of  a  given  surface  at  a  given  point. 

Assume  the  tangent  plane  at  the  given  point  as  Ibtt  of  xy\  the 
normal  coinciding  with  the  axis  of  z. 


CURVATURE   OF  SURFACES. 


237 


Let  OX^  be  the  trace  of  the  se- 
cant plane  on  that  ol  xy^  forming 
with  OX  an  angle  1  AOB  the 
normal  section,  and  P  a  point  in 
that  section.  Put 
OEz^x,  ED-y,  DP—z,  OD=x^ 
The  co-ordinates  of  the  curve 
AOB,  estimated  in  its  own  plane, 
are  x^  and  z ;  and  the  general  value 
of  the  radius  of  curvature  of  a  plane  curve  where  x-^  and  z  are  the  co- 
ordinates, and  any  quantity  t  the  independent  variable,  is  (Art.  131."^ 

3 


Ez= 


K] 


'dfi'  ~di 


d'^x-^   dz 
If '  di 


which,  applied  to  the  present  case,  making  t  =  ar,  and  observing  that 


-    ds       dx.        ^    dz        .        ^ 
at  0  —r~  =  —r^  and  -r-  =  0,  reduces  to 


dx       dx 


dx 


R  = 


m 


(Pz^ 
dx^ 


(!)■ 


dH 


Jn  this  expression,  the  coefficient  -^  has  reference  to  those  points 

of  the  surface  which  lie  in  the  curve  A  OB^  and  therefore  it  differs 

d'^z 
from  the  partial  differential  coefficient  -j-^  derived  from  the  equation 

of  the  surface,  which  latter  refers  to  the  change  in  z  produced  by  a 
change  in  x  only,  while  y  is  constant. 

Let  z  z=i  (p(ir,3/)  be  the  equation  of  the  surface  ;  then     (Art.  55) 

dy 


[dz~\  ^dz 
dxA  ~  dx 


»ent  case. 


dz    dy      dz       dz 
dy    dx~  dx       dy 


snice  y-  =  tan  &  in  the  pre- 


*238  DIFFEEENTIAL   CALCULUS. 


r-i= 


d^z       ^    d^z  ,        d^z       ,, 

dx^  dxdy  dy^ 


dx            1 
Also  —7-^  = T*     Hence  by  substitution  in  (1)  and  reduction, 

c&»C/  COS  u 

^"^l^z        ~       ~~d^        ~7~;        dH      ~7  '  '  *  ^^^' 

——  cos^d  -f-  2  -J— r-  cos  B  .  sni  &  -\ — ;-r-  •  sin-^^ 
dx^  dxdy  dy^ 

205.  Prop.  The  sum  of  the  curvatures  of  any  two  normal  sec- 
tions of  a  curved  surface,  drawn  through  the  same  point  of  the 
surface,  and  perpendicular  to  each  other,  is  constant,  those  curva- 
tures being  measured  by  the  reciprocals  of  the  radii  of  curvature. 

Let  &  and  &^  be  the  inclinations  of  the  secant  planes  to  the  plane 
{)ixz\  R  and  R^  the  radii  of  curvature  of  the  two  sections  at  theii 
common  point.  Then,  since  the  sections  are  perpendicular  to  each 
other, 

^1  =:  -  -TT  -f-  ^,     and     .  • .    cos  4  =  sin  ^j,     sin  4  =  —  cos  d^, 

and  by  formula  [P] 

1        d'^z        ^,   ,    ^  d^z  ,   .    ,    ,   dH     .  ^, 

-7-  =  -7-^  •  cos2^  +  2  -—r  •  cos  ^  sm  ^  +  -r-s  •  sni^^. 
R       dx^  dxdy  dy^ 

1         d'^z  d^z  d^z 

Hence  by  addition  and  reduction 

\         \        d'^z      dH  ^       u 

--  4-  -TT-  =  -7—-  +  -7-7:  =  a  constant  lor  the  same  pomt. 
R      i?i       dx^       dy^ 

Cor.  The  normal  sections  of  greatest  and  least  curvature  at  anjF 
point  of  a  curved  surface,  are  perpendicular  to  each  other. 

For  since  -—  -j-  -—  is  constant,  -r-  will  be  greatest  when  ~  is 
R       Ri  R  /S| 

least,  and  it  will  be  least  when  -^  is  greatest. 

^1 


CURVATUEE  OP  SURFACES.  '^9  ?', 

206.  The   sections   of  greatest   and   least   curvature   are   called     '  *  \  * , 
principal  aections^  and  the  corresponding  radii  are  called  principal  -"f 
radii.                                                                                                                           ^""^w^ 

207.  Prop.  To  determine  the  principal  radii  of  curvature  at  a 
given  point  of  a  curved  surface. 

By  differentiating  —  with  respect  to  ^,  as  an  independent  variable, 

and  placing  the  differential  coefficient  equal  to  zero,  we  get 

H  (Pz  (Pz 

d?z 

+  2  -r^  •  sin  a  cos  ^  =  0. 

d'^z  Td^z      d^zl_   d^z 

dxdy  Ldy^       dx^J       dxdy  ^    '' 

From  which  we  obtain  two  values  of  cot  L  viz. : 


_  ^2  ~  ^2  ^  V  \jf  ~  d^U   "^     \d^y} 
CO       _  —^ 

dxdy 

Substituting  this  value  in  the  formula  (P),  which  may  be  written 
thus 

1  +  cot2^ 


R 


d^z  10^^^       f  4  _L  ^^^ 

dz^  dxdy  dy^ 


and  denoting  by  i?i  and  i?2  the  least  and  greatest  radii  of  curvature, 
there  results 

Ri=- ; =zl==== (B). 

^2^  -JV-.  /n    J9-.  79.-19  y       J9         VO  V         / 


f;x2 


i23=: : (S). 

dx-'  "^  rfj/2       V  L^^2      ^^2j  +  "^Xdxdy) 


240  DIFFERENTIAL   CALCULUS. 

208.  Prop.  To  express  the  radius  of  curvature  of  any  normal 
section  in  terms  of  the  principal  radii  R^  and  i?25  ^^^  ^^^  angle  9 
formed  by  that  section  with  the  principal  section  of  greatest  curva- 
ture. 

If  we  make  successively  ^  =  0,  and  &  =  -ic  in  [P]  we  obtain 

tit 

And  these  will  be  the  values  of  R-^  and  ^^,  if  the  planes  of  xz  and 
yz  be  supposed  to  coincide  with  those  of  greatest  and  least  curva- 
ture.    Thus  we  shall  have,  upon  this  supposition, 

dx^  ~  R;     ^^       dy-^  ~  R^ 

d'^z 
The  same  supposition  renders  =z  0,  as   appears   when   we 

put^  =  0  in  [Q). 

These  conditions  reduce  (P),  when  &  is  replaced  by  (p,  to  the  form 

jn  R\tlz r  yr-j 

~  R^Q.ii^^^  -j-  Z^^sin^^p  ^     ^' 

the  desired  formula. 

209.  Prof.  If  the  two  principal  sections  of  a  curved  surface,  at 
any  point,  have  their  concavities  turned  in  the  same  direction,  then 
every  normal  section  through  that  point  will  be  concave  in  the  same 
direction. 

In  the  formula  (7"),  the  signs  of  i^^and  R.^  depend  upon  those  of 

d'^z  d'^z 

-T-T7  and  -1—:;  and  the  signs  of  these  coefficients  indicate  the  direc- 

dx^  dy^  '  ^ 

tions  of  the  curvature  of  the  principal  sections. 

In  the  case  under  consideration,  the  signs  of  R-^  and  i?2  ^^^"st  be 
alike,  arid  therefore  if  both  be  +,  the  sign  of  R  will  be  -|-  also; 
but  if  \n)\\\  1)0  — ,  then  the  sign  of  R  will  likewise  be  negative. 

Froii!  which  the  truth  of  the  proposition  is  apparent. 


CURVATURE  OF  SURFACES.  241 

)>lv.  Cor.  If  i?i  and  i?2  be  also  equal,  then  R  =  E^  =  B2  for 
w<ery  vdlne  cf  f),  and  every  normal  section,  through  the  same  point, 
will  hi  ve  tl^e  same  curvature.  This  occurs  at  the  vertices  of  sur- 
faces of  revolution, 

211.  Prop.  If  one  principal  section  of  a  surface  be  concave,  and 
the  other  convex,  it  ^\ill  be  possible  to  select  a  value  (p-^  for  9,  which 
shall  render  E  infinite,  or  the  section  a  straight  line ;  also,  between 
the  values  9  =  —  9^  and  :p  -  -l-  (pj,  the  signs  of  E  and  E^  will  be 
alike ;  but  from  9  =  (pj  to  9  ■=-■  •n'  —  9^,  the  signs  of  R  and  E2  will 
be  alike. 

In  the  formula  [7^],  suppose  E^  negative,  and  it  will  become 


E  = 


B,R, 


/tjcos^vp  —  M^  s)n29 
in  which  transformed  expression,  the  quaniities  R-^^  and  E2  are  to  be 
<X)nsidered  essentially  positive. 

Now  suppose  9  so  taken  that  E^  COS29  —  y?^  3ia*9  -=  0,  a  condition 
that  will  be  fulfilled  when 


=  9i  =  tan- 


m  "•  —'-m. 


Then  i2=,ZlAj??^c^. 

Thus  there  are  two  sections  corresponding  to  the  argles  f  ^  and 
-  9|  which   give   straight   lines.     Also,  if  9>  — 9^  and  9<9i; 
then  E2  cos2  9— i^j  sin29  >  0,  and  .  • .  72  <  0. 

But  if  9  >  9i  and  9  <  *  —  9i,  then  E^  co&^cp  —  Ri  sin29  <  0, 
and  i2  >  0. 

Hence  the  surface  may  be  divided  into  four  parts  by  two  planes, 
and  if  the  first  of  these  parts  be  supposed  concave  the  second  will 
be  convex,  the  third  concave  and  the  fourth  convex. 

212.  Prop.  To  determine  whether  the  principal  radii  at  any  point 
have  the  same  or  contrary  signs,  the  co-ordinate  planes  not  being 
coincident  with  the  principal  sections. 

16 


242  DIFFERENTIAL    CALCULUS. 

The  general  values  of  R^  and  i?2  ^^^Y  ^®  reduced  to  the  forms 

R2  = 


p"  +  q"  +  vW+^"f  -  ^pW  -  «"') 

2 

P['  +  9"  -ViP"  +  9"Y  -  ^{P"9"  -  ^' 


and  these  values  will  have  the  same  sign  when  'p"q"  —  s"'*^  >  0,  and 
contrary  signs  when  "p^'q"  —  s"^  <  0. 

213.  Prop.  At  every  point  of  a  curved  surface,  a  paraboloid 
(either  elliptical  or  hyperbolic)  can  be  applied,  with  its  vertex  at 
that  point,  which  shall  have  contact  of  the  second  order  with  the 
given  surface. 

Assume  the  point  of  contact  as  the  origin,  the  normal  being  taken 
as  the  axis  of  2;,  and  the  planes  of  xz  and  yz  coincident  with  the 
principal  sections  of  the  surface. 

Take  the  normal  as  the  axis  of  the  paraboloid,  its  vertex  being  at 
tbe  point  of  contact,  and  turn  the  paraboloid  about  its  axis  until  iU 
principal  sections  coincide  with  xz  and  yz.  The  equation  of  the 
paraboloid  when  in  this  position  will  be  Ax^  ±  By"^  =  Cz^ 

which  may  be  written       z  =  ~j^  rb  —— ? 

c  c 

where  2P  =  —  and  2Pi  =  ~,  which  represent  the  parameters 
of  the  principal  sections,  are  entirely  arbitrary. 

Take       P  =  i?j,     and     P^  =  R^.     Then     2  =  ^^  ±  ^• 

„  rf22  1  c?22  1 

and  therefore  R-^  and  J?,  are  the  principal  radii  of  curvature  of  the 
paraboloid  also.     Then,  for  any  other  normal  section  of  the  parabo- 


CURVATURE   OF  SURFACES. 


243 


loid,  we  shall  have  R 


±i2A 


the  same  value  as  that 


i?i  sin2(p  ±  R2  cos2(p' 
of  the  radius  of  curvature  of  the  corresponding  normal  section  of  the 
surface.     (Art.  208). 

Cor.  It  appears  that  when  the  principal  sections  of  two  tangent 
surfaces  have  contact  of  the  second  order,  every  other  normal  sec- 
tion made  by  the  same  plane  drawn  through  the  same  point  will 
likewise  have  contact  of  the  second  order. 

214.  Prop.  To  determine  the  radius  of  curvature  of  an  oblique 
section  of  a  curved  surface. 

Take  the  point  of  contact  as  the  origin,  and  the  tangent  plane  as 
that  of  xy. 


Let  OXj  be  the  trace  of  the  secant  plane  on  a;y,  aOb  the  section 
of  the  surface  by  that  plane,  A  OB  the  normal  section  by  the  plane 
ZOX,  R  the  radhis  of  curvature  of  AOB  at  0,  r  the  radius  of 
curvature  of  aOh  at  0.  Draw  OZ^  perpendicular  to  OXj,  in  the 
plane  a  Ob,  and  refer  that  section  to  the  rectangular  axes  OXj  and  OZj. 

Put  Od  '—  ar^,  dp  =  z-^,  X  =  angle  between  a  Ob  and  A  OB^ 
pD  ^z,   DE=  y,   OE  =  x. 


244  DIFFERENTIAL    CALCULUS. 

Then  at  the  point  0  we  shall  have 

l.dxj  LdxJ 

dx^  dx^ 

d?-z      d^z,  ^         . ,  ds-,       dx-i       ds 

But     0  =  0,  cos  X.     .  • .  -7-:  =  -r-^-  •  cos  X.      Also      —  z=-—-  =  -—> 
*  dx^      dx^  dx        dx       dx 

.'.  r  =  E ,cos\ 

^nd  consequently  radius  of  the  oblique  section  =  projection  of  th« 
radius  of  the  normal  section,  on  the  plane  of  the  oblique  section. 
This  result  is  known  as  Meusnier''s  Theorem. 

Cor.  If  a  sphere  be  described  whose  radius  shall  be  identical  with 
that  of  the  normal  section,  and  if  through  the  tangent  to  that  section 
any  plane  be  drawn  intersecting  the  sphere  and  the  given  surface, 
then  will  the  small  circle  cut  from  the  sphere  be  osculatory  to  the 
fwrve  cut  from  the  surface. 

Lines  of  Curvature, 

215.  If,  through  the  consecutive  points  of  any  curve  traced  upon 
a  given  surface,  normals  to  that  surface  be  drawn,  such  consecutive 
normals  will  not  usually  lie  in  the  same  plane,  and  therefore  will 
not  intersect;  but  when  the  consecutive  normals  do  inteij^ect,  the 
corresponding  curves  (which  enjoy  peculiar  properties)  are  called 
lines  of  curvature. 

216.  Prop.  To  determine  the  lines  of  curvature  passing  through 
any  point  on  a  curved  surface. 

Let  the  equations  of   the   normals  passing  through   any  point 
(*!'  y\^  2;i),  be 
T-x^-^t{z-z;)=0=P  . .  .{\)  and  y-y^^s{z-z;)=0=Q  . .  .(2), 

and  suppose  the  independent  variables  x  and  y  to  receive  the  incre- 
ments h  and  k. 


LINES  OF  CURVATURE.  245 

Then  the  equations  of  the  normal  in  the  new  position  will  be 

<i+S-'-=» «• 

U  these  two  normals  intersect,  the  equations  (1),  (2),  (3),  and  (4)^ 
will    apply  to  the   point  of  intersection;    and   if  the  co-ordinatf 
K,  y,  and  s  of  that  point  be  eliminated  between  the  four  equations, 
the  result  will  be  a  relation  between  the  increments  h  and  k  and  con- 
stants, it  being  observed  that  t  =  — ^  and  s  =  -r^,  are  constant  foi 

^dP    dP    ^ 
the  same  pomt,  and  the  same  is  true  oi  -7—,  -7—,  &c. 
^  dx^  dy^ 

This  relation  between  h  and  k  implies  a  necessary  relation  between 

the  new  values  of  x  and  y,  in  order  that  an  intersection  of  the  nor 

mals   may   be   possible ;    and   when    the  normals   are  consecutive, 

A  =r  0,  and  ^  —  0,  and  -  =  -j^     Thus  by  omitting  P  and   Q  (each 

of  which  is  equal  to  zero)  in  (3)  and  (4),  then  dividing  by  A,  ant' 
finally  making  A  =:  0,  those  equations  become 

dP         dP_   dy^  .      dQ       dQ   dy^  _ 

dx-^         dy^    dx^  ~      '  *  *  *  ^  -''  ^^^       ^y^   dx^  ~      *  *  *  v  /> 

or,  by  forming  the  values  of  the  partial  differential  coefficients, 
dP     dP    dQ        ,  dQ   ^        .,.       .  .^. 


^  dx^     dx^     ^        ^'dx-^dy-^   dx-^     dxj    dy^    dxi       ' 

^^^(z-z\-^    ^^1     ^yi  I  /^     ^  ^d%   dy^_dzl   ^i-O- 
dx^dy^        '^     dy^'dx^     c/a:i     ^         ^Uy^^' dx^     dy^' di~    \ 

and  by  eliminating  z  —  0^,  putting 


(7), 


246  DIFFEKENTIAL   CALCULUS, 

we  obtain 

J;-,[*"(i+i?'^)-y?'?"^] 
+  ^[p"(i+?'^)-?"(i+y^)]-s"(i+/^)+/?y'=o...(i/), 

dv 
This  is  a  quadratic  equation,  giving  two  values  of  -^,  the  tangent 

of  the  angle  between  the  axis  of  x  and  the  projection  of  the  tangent 
to  the  line  of  curvature  passing  through  (arj^/i^i),  upon  the  plane 
of  xy.  Hence  there  will  be  two  lines  of  curvature  passing  through 
mch  point  of  the  surface ;  and  if  jc>',  q\  &c.,  be  replaced  in  (f^)  by 
ilieir  general  values  derived  from  the  equation  of  the  surface,  the 
result  will  be  the  differential  equation  of  the  projection  of  every 
pair  of  lines  of  curvature  upon  the  plane  of  xy. 

217.  Prop.  The  lines  of  curvature  at  any  point  of  a  curved  sur- 
face intersect  each  other  at  right  angles,  and  they  are  respectively 
tangent  to  the  sections  of  greatest  and  least  curvature. 

If  we  suppose  the  plane  of  a??/,  (which  in  the  last  proposition  was 
assumed  arbitrarily)  to  coincide  with  the  tangent  plane  at  the  point 
tuider  consideration,  we  shall  have 

y=|^=0,     and     /  =  |;-  =  0. 

Hence  the  equation  ( CT)  may  be  reduced  to  the  form 

^    y^^^_         

dx^^  ^     y      dx^  ^   ^ 

Hence  if  ^j  and  ^2  denote  two  angles  determined  by  the  condition 
that  tan  ^^  and  tan  ^2  shall  be  the  roots  of  this  equation,  we  shall 
have,  by  the  theory  of  equations, 

tan  dj  tan  dg  =  —  1,     or     1  +  tan  ^^  tan  ^g  =  0, 

which  is  the  condition  of  perpendicularity  of  two  lines  in  the  plane 
of  xy  forming  angles  dj  and  ^2  with  the  axis  of  x.  Thus  the  tan- 
gents  to  the  two  lines  of  curvature  intersect  at  right  angles. 


LINES    OF   CURVATURE.  247 

218.  Again,  if  we  divide  equation  (  F)  by  -~^  =  tan^^  and  replace 

by  cot^,  the  result  will  become  identical  in  form  with  equation 

(Q),  which  serves  to  determine  the  two  angles  formed  by  the  prin- 
cipal sections  with  the  plane  of  xz^  and  hence  the  directions  of  the 
lines  of  curvature  are  tangent  to  the  curves  of  principal  section. 

219.  Prop.  The  consecutive  normals  to  a  surface  drawn  through 
points  in  the  lines  of  curvature,  intersect  at  the  same  points  as  the 
consecutive  normals  to  the  principal  sections  to  which  the  lines  of 
curvature  are  tangent. 

Regarding  the  tangent  plane  at  the  given  point  of  the  surface  as 
still  coincident  with  that  of  a;y,  we  shall  have 

dz  dz 

z^  =  0,    -—  =  0  and  -~-  =  0  and  the  equation  (7),  gives 
U/X-I  Vx 

1  tand 

or 


dx-^       dx-^dy-^  dx^dy^       dy-^ 

Now  if  the  plane  of  xz  be  supposed  coincident  with  a  principal 

d'^z 
section,  these  expressions  will  be  still  further  simplified,  since       ^ 

will  then  be  =  0 ;  thus, 

__1_  _    1 

d^z-^  ~  d'^z-^ 

dx-^  dy-^ 

But  these  expressions  are  precisely  the  same  as  those  previously 
found  for  the  radii  of  curvature  of  the  principal  sections,  and  henco 
the  centres  of  curvature  of  the  principal  sections  must  coincide  with 
the  points  of  intersection  of  consecutive  normals  to  the  surface 
through  points  in  the  lines  of  curvature. 


INTEGEAL   CALCULUS.    PART   L 


CHAPTER    1. 


FIRST    PRINCIPLES. 


1.  The  ©bject  of  the  Integral  Calculus  is  to  determine  the  function 
from  which  any  proposed  differential  has  been  obtaim-d.  The  pro- 
cess by  which  this  is  effected  is  called  integration^  and  is  indicated 
by  the  sign  /,  the  result  being  called  the  integral  of  the  proposed 
differential. 

2.  Whenever  the  given  differential  can  be  reduced  to  a  knowi 
form,  we  may  return  to  the  function  by  simply  reversing  the  rulea 
for  differentiation. 

3.  Since     d(a.Fx)  3  a  .  d(Fx)  =z  aFyX.  dx,  we  infer  that 

faFyX  .dx  =  a  f  F^x  .  dx^ 

that  is,  we  may  remove  any  constant  factor  from  under  the  sign  of 
integration,  placing  it  as  a  factor  exterior  to  that  sign. 

/a  1 

-  F^x  .dxz=z-Ja.  F^x .  dx. 

Therefore  we  may  introduce  a  constant  factor  under  the  integral 
sign,  provided  we  write  its  reciprocal,  as  a  factor,  exterior  to  that 
sign. 

5.  To  differentiate  the  algebraic  sum  of  several  functions,  we  dif- 
ferentiate each  function  separately,  and  take  the  algebraic  sum  of  the 


ALGEBRAIC  FUNCTIONS.  249 

several  differentials.  Hence,  in  order  to  integrate  the  algebraic  sum 
of  several  differentials,  we  have  only  to  integrate  the  several  terms 
successively. 

Thus     f{adx  -f  hdy  —  cdz  +  edv)  =  fadx  -\-fhdy  —fcdz  -\-fedv 

z=z  ax  +  by  —  cz  -\-  ev. 

6.  Again,  since  differentiation  causes  all  constants  connected  with 

the  variables  by  the  signs  +  and  —  to  disappear,  it  follows,  that  in 

effecting  an  integration,  we  should  always  add  a  constant,  in  order 

to  provide  for  that  which  may  have  disappeared  by  differentiation : 

thus  we  write 

fadx  z=  ax  -{-  c^ 

in  which  the  value  of  c  will  be  arbitrary,  unless  fixed  by  other  con- 
ditions. 

Suppose,  for  example,  that  the  general  value  of  the  integral  is  JT, 

so  that 

X  =:  ax  -\-  c\ 

and  that  for  a  particular  value  x^  of  x^  the  integral  assumes  a  known 
value  Xj:  then 

Xj  =  ax-^  +  c,  and  .  * .  c  =  Xj  —ax^. 
And  this  value  substituted  in  the  general  integral,  gives 
X  z=z  a{x  -  x^)  +  X^. 

Integration  of  the  Form  (Fx)°dFx. 

7.  Prop.  To  integrate  the  form   {FxYdFx. 

Here  we  have  /{Fx^dFx  =  f{n  +  l){Fx)"dFx 

=  -^  fd{FxY+^  =  !^^^.  e. 
n  -\-  I       ^      [  w  -f-  1 

The  same  process  can  obviously  be  applied,  whenever  the  quan. 
tity  exterior  to   the  parenthesis,   can   be   rendered    the  exact  dif- 


250  INTEGRAL    CALCULUS. 

ferential  of  that  within,  by  the  introduction  or  suppression  of  a 
constant. 

Hence  we  have  the  following  rule  for  the  integration  of  this  form, 
viz. :  ' 

Divide  the  given  expression  by  the  differential  of  the  quantity  within 
the  (  ),  then  increase  the  exponent  of  the  (  )  by  unity ^  and  finally^ 
divide  by  the  exponent  thus  increased. 

EXAMPLES. 

8.  1.  To  integrate  ax^dx. 

a  ax* 

faxHx  =  a  fx^dx  —  -fAx^dxz=—  -f-  c. 

2.  To  integrate  -y^^^  -f  x^ .  ZcxHx. 

f{b^  -f  a:*)*  Scx^dx  =^'l  f  1(62  +  a:*)*,  ^^'dx  =  ^(b^  +  x^)^  +  C 

3.  To  integrate  dy  —  (2a  +  Sbxydx. 
This  may  be  integrated  in  two  ways ;  thus 

y  =  f(2a  +  Uxfdx  =f(Sa^  +  SiJa^x  +  64ab^x^  +  21bH^)dx 

=  fSaMx  4-  f'SCm^xdx  +  fh4ab^x^dx  +  f^lb^xMx 

27 
=  ^a^x  +  18a26a;2  +  l^abH'^  +  —■  ¥x^  +  c (1). 

Again 

y  =  /(2a  4-  Zbxfdx  =^  /4(2a  +  36a:)3.  Udx  =  ^(2a+362:)*+c- 

4a>                                                          27 
=  ot4-  8a3a:  4-  18a26a;2  +  18ai2a;3  -f  —63a;*  +  Cj (2). 

The  formulae  (1)  and   (2)  are  identical.     For  if  yj  denote  the 

particular  value  of  y  when  a;  =  0,  we  shall  have  from  (1)     yj  =  c ; 

4a*  4a* 

and  from  (2)     y^  =  —  +  c^,     • '  •  ^  =  3^  +  ^i- 


ALGEBRAIC  FUNCTIONS.  251 

4.  To  integrate    dy  =  3(46a:2  _  2cx^y  {4bx  —  Scx^)dx 

y  =  I  /(46a;2  —  2czY  {^^^  —  Qcx^)dx  =  |  {4bx^  -  2cz^y  +  c. 

9.  In  each  of  the  preceding  examples  the  proposed  differential  has 
Deen  brought  to  the  required  form,  viz. :  that  in  which  the  part  ex- 
terior to  the  (  )  is  the  exact  differential  of  that  within,  by  intro- 
ducing a  constant  factor.  To  ascertain  when  this  is  possible,  take 
the  last  example,  and  denote  by  A  the  required  unknown  factor :  then 

y  =  \f(Ux^  -  2cx^)^(12Abx  -  9Acx^)dx, 

and  if  this  be  of  the  required  form,  we  must  have 

d(Ux^  —  2cx^)  =  (12Abx  —  9Acx^)dx 

or  Sbx  —  6cx^  =  \2Abx  —  ^Acx"^, 

and  since  this  condition  must  be  satisfied  without  reference  to  tlie 
value  of  X,  we  must  have,  by  the  principle  of  indeterminate  coeffi. 
cients,  the  two  separate  conditions 

86  =  \2Ab (1)     and     -  6c  =  -  Mc (2). 

Fro.n(l)      ^=:^=?      and  from  (2)     ^=|  =  | 

The  values  of  ^  derived  from  (1)  and  (2)  being  identical,  the  pro- 
posed reduction  is  possible. 

The  next  example  will  illustrate  the  contrary  case. 

1.  dy  =  {Ab-^x  -f  3aa;2)^(262  +  %ax)dx. 

If  possible,  let  A  be  the  required  factor.     Then 

y  =  l.f(^b^x  +  ^ax^y  (2b^A  +  SaAx)dx, 
A 

and  .  • .  d{4b^x  -f  Saz'^)  =  (26M  +  SaAx)dx, 

or  462  ^  Qax  =  262^  -f  SaAx, 


262  INTEGRAL  CALCULUS. 

which  gives  the  two  separate  conditions 

462  _  262^ (1)         and         Qa  =  SaA (2). 

From(l)      A  =  ^^=z2,     and  from  (2)     A=~  =  ^^ 

These  values  of  A  being  different,  the  desired  reduction  is  impossible. 

2.  10  mtegrate  ay  =  ;:—^» 

adx 

3,  dy  = 


X  y/ohx  -\-  '^C^X^ 

i 


y  =  afx~\Sbx  +  ^c^x^f^dx  =  ~f{3bx-^  +  4c2)  *.  Sbx-^dx 

oO 


axdx 
4.  rfy  = 


{2bx  +  a;2  * 


y  =  af  {2bx  +  a:2)"'.  a^^/i:  =  a  f(2br-^  +  1  )~^  i^"^)      •  ^dz 
^  l.f  (2bx-'  +  1  )~"^.  2;-2 .  2bdx  =  ^  (26ar-i  -f  1)~*4.  c 
a  r2bx  +  ar2"|— i    ,  ax 

=  r    5 —  \    ^  -\-  c  =  — +  c. 

bL      x^       J  h^2bx  +  a:2 

5.  dy  =  —^ ^  dx. 

X  —  a 

y  =  3/a;*(a;2  +  ax  +  a2)af2;  =  3  /  (a:0  4-  aa:«  +  a^x^)dx 

+  c. 


/x-'       ax^       a'^x^\ 


CHAPTER    II. 

iLBMENTARY   TRANSCENDENTAL   FORMS. 

Loga/rithmiG  Forms, 
10.  Prop,  To  integrate  the  forms  and  — ^ — '— 

X  JuX 

Since  d{a  log  x)  = .',  J  =  a  log  x  -\-  c. 

T/     1      T^  X       a.dFx  Pa.dFx  .      „    . 

Also  since  d{a .  log  Fx)  =  — — —     .  • .  /  — = —  =  a .  log  ^a?  +  e. 


EXAMPLES. 


adx 
11.  1.  To  integrate  dy  =  ,    .    '  » 

°  b  -^  ex 


^'=tfb^x=^^^'^^''-^''^  +  ^  =  ^"^  [(^+  ^^)^]  +  (^' 


2.   io  integrate  ay  = 


a  -h  2^ 
^==y  ^2^=^^S(«+2;r*)+  (7=:log(a+2:r*)  +  logc=log[c(a4-2x*)]. 

In  this  example  the  constant  introduced  by  the  integration  is  put 
into  the  form  of  a  logarithm  (which  is  always  admissible)  for  the 
purpose  of  simplifying  the  form  to  which  the  integral  is  finally 
reduced. 


264  INTEGRAL  CALCULUS. 

'7xdx 
3.  To  integrate  a 


8«  -  Sx^ 


1                        c 
=  logc  —  log  (8a  —  Zx^y  =  log 


(8a  -  3a;2)* 


4.  io  intepjrate         a?/  =  — ^ ^^^ 


y 


_  b_  r{S\x^  -  lOSx^a^  +  b4x^a^  -  12xa^  +  a^)dx 
n*/  or? 


or 


,      ,^_/L81.-108a3+_-_  +  ^J^. 

:^  ^["^  a;2  -  108a2:r  +  54a*  log  a:  +  —  -  -^1  +  C7. 

Circular  Forms. 

dx 
12.  Pro/?.  To  integrate  the  form     dy  =z  ±.  

Taking  the  upper  sign,  we  have 

I         -^^  1  1         -^^ 

4-  ««  I         «  1  I         a 


P— ——-——-    —  I —   -  I    -- - 


Let  the  quantity  under  the  sign  of  integration  be  compared  with 

dz 
the  well  known  form  c?(sin-"^2)  =  —         -,  and   it   will   be   found 

identical  therewith,  provided  we  make  -x  =.  z. 

r  -'- 

But    /  — : =  sm-^0  +  c,     .  •.     I  —  =sm~^  —  -tc 

^        6  a 


a 


TRANSCENDENTAL  FUNCTIONS. 


255 


Similarly,  since    /  -  =  cos""^^  +  c. 

—  dx           1            ,  &^   . 
—  =  -  •  cos-i \-  c. 


dx 


13.  Prop.  To  integrate  the  form  dy  =  ±    ^  ""  „    « 
Taking  the  upper  sign,  we  have 


r  1 


'=/. 


dx 

2  4-^2^2 


c?:c 


1  + 


/j2;i;2         ab 


-dx 
a 


Comparing  the  expression  under  the  sign  of  integration  with  the 

dz 


well  known  form    c?(tan~^2;) 

,.        bx 
inakmg    —  =z  z. 


l+z 


-,   they   become    identical    by 


But 


/rf72  =  tan-i.  +  c.    .-. 


1  + 


-  dx  , 

— 7?-o  =  tan-i-a;4-<?. 
o2a;2  a 


.  • .  y  z=:—- tan~i h  <^« 


1^  =  COt-^2  -{-  C. 

•••  y-f- 

t/  a 


2  -L  ^2a;2  „5  d 


14.  Prop.  To  integrate  the  form  dy  =i  dtz 


dx 


Taking  the  upper  sign,  we  have 


aryPi 


-dx 


dx 


•256 


INTEGRAL   CALCULUS. 


Comparing  the  expression  under  the  sign  of  integration  with  the 

dz 

known  form  o?(sec"~^2;)  = ,  they  become  identical  by  making 

z^^/z^  —  1 
hx 
a  f  0 

But  f — ; =  sec-^z  +  c.   .  • 

J zJ7-l 


dx 


hx     Ih^x^ 


secri J-c. 


1       -^hx  . 

• .  2/  =  -  sec \-  c. 

a  a 

—  dz 


And  similarly,  since    / =  cosec'^g  +  <?• 

J  Zyfz^  -\ 


•/    X 


-dx 


y/hH''  -  Q? 


=  -  cosec" 


bx 


+  c. 


dx 


15.  Prop.  To  integrate  the  form  c?y  =  ± • 

ya^x  —  b'^x^ 
Taking  the  upper  sign,  we  have 

r       2b 


/+  dx 


dx 


UbH      46*^2 
V  ~a2  ^ 


262 


dx 


J 


V  "^      ~^ 


262 


dx 


H^-)-m 


Comparing  the  expression  under  the  sign  of  integration  with  the 


known  form   cif(versin~^^)  = 


dz 


'^z  —  z^ 


,  they  become  identical   by 


2^2.c 

making   — —  ■=.  z. 

dz 

y/2z  —  Z' 

r         262 


But 


— ^==  =  versin-^2r  -f-  c. 

^/22  —  ^2 

dx 


J 


vM*)-(W 


=  versm-*  — —  -f  c, 
a2 


TRANSCENDENTAL  FUNCTIONS.  257 


1         .     ,  ^b'^x 


.   y  =  -l  versin-i  — ^  -f  e. 
0  a/' 


—  dz 
And  similarly,  since    /  —         —  =  coversin-^0. 


dmilarly,  since    f—i^ 


EXAMPLES.  \-^^4.. 


16.  1.  To  integrate  dy  = 


■x/a^  -  62^* 


y-¥b 


^    '^  dx 

a  1      .     ,  bx^ 

:=  =  —  sm-1  —  -f  c. 
b^x*      26  a 


J 


xA^ 


x^dx 
2.  To  integrate  dy  =  j-q_— g' 


1    fSx^dx         1  w  3x    , 


-f 
^    _     .  -  Sx     dx 

3.  To  integrate  ay 


-i  C  -I 

/-I  8a;  *c?a;  .    r^\  2x   '  dx 

y  =  yb 


J  2  .Qx^  -Q.Qo^  L/2  .  62;*  -  6  .  6x* 

=  4  ye .  versin~i(6a;')+  c. 

17.  Since  each  of  the  trigonometrical  functions  can  be  expressed 
in  terms  of  any  other,  all  the  circular  forms  must  apply,  whenever 
one  is  applicable.     To  illustrate  this,  take  the  example 

or  dx 


dy  z=  — - — 

■x/2^4x^ 


17 


268 


y  = 


INTEGRAL   CALCULUS. 


—--xdx  f  i 


'3  It 

-^l.x   dx 
2^  1 


-v/l  -  2a:3       3 


=  oSm-*V^^-f<^. 


J 


r»         Q  1 


or  y  = 


^^____      =.leos-V^  +  c, 


J 


Again, 

/»      x'^dx  1    /> 

1    /.      -  12a;2c^ 
^y  n/2 .  4:r3  - 


or,  2/ 


:r —  =  -  versm-i  [Ax^)  -f-  c« , 

4a;3_(4a;3j2  6 


1 


^  ,      , =  =  —  ^  coversm-i  (4a:3)  +  e^, 

^J  V2.4:r3- (42:3)2  6  V      /T-   3 


Again,  y  =  y- 


ic  a:     dx 


\s/l^'^dx 


-2\/2'"^"        1    ,  /r 

V2-"    Vr  -^ 


a?    4-«i 


or 


1 

y  =  -  cosec- 


3 


X    4-  c«. 


a:-3_l|    *c/a; 


Finally,  y  =  p-x^x^i^^x-^-\^ 
=  -itan-y| 


^  -  1  +  «?. 


01 


=  -  cotan-y  -  ar-3—  1  4-  <v 


TRIGONOMETRICAL  FORMS.  259 

Trigonometrical  Forms. 

18.    Prop.    To    integrate   the    forms    sin  xdx^  cos  x  dx,  seG^xdx^ 
Qosec^xdx,  sec  x  tan  xdx^  and  cosec  x  cot  xdx. 
Since d(cosx)=z—sinxdx^ .'.  fsmxdx=z  —f—smxdxz= — cosic-f-c. 

"    c?(sin  x)  =     cos  xdx, .  ' .  /cos  xdx  =.  sin  x  -\-  c. 

"    c?(tan  x)z=     sed^xdx^  .  • .  fsec^xdx  =  tan  x  +  c. 

"    c?(cot  x)=z  — cosea^xdx,  .  * .  fcosed^xdx  =  —  cot  x  -{-  c. 

"    c?(sec  x)  =     sec  x  tan  irc^ar,  .  • .  /sec  re  tan  xdx  =  sec  a:  +  c. 

"  6?(coseca:)  =  — cosec ircotaro?^;, .  *. /cosec a*cota;G?a;=  — coseca;-f-<?. 

EXAMPLES. 

19.  1.  To  integrate  dy  =  2  cos  ^x .  dx, 

2  /*  2 

y  =  /2  cos  8a; .  6?a;  =  -  /  cos  Sx .  d(Sx)  =  -  sin  Sx  -f  c. 

2.  fl?y  =  5  sec^  (a;^) .  ar^c^. 

y=:/5  sec2(a:3) .  a;2fl?a:=|ysec2(a;3)  3  a;2c/a;=|y"sec2(a;3)  (/(a:3) 


3 

dy  =  6  sec  4x .  tan  4a: .  dx 


5 

=  -tan(a;3)-}-c. 


6   /•  3 

y  =.-  I  sec  4a: .  tan  4a: .  c?(4a:)  =:  -  sec  Ax  +  c, 

dy  =  2  sin  (a  -f-  3a;)c?a:, 
y  =  ^J  sin  (a  +  3a:)  Sdx  =  ~J  sin  (a  +  3a:) .  c?(a  -{-  3a:) 


2 

=  —  -  cos  (a  -|-  3a;)  +  c, 
o 


3  1 

5.  6?y  =  -cosec2(-y/2^  .  a:^c^a:. 

V  =  --^y*cosec2(v^) . -y^.  a:  Va;  =  -  -y=  cot y^  + 


260  INTEGRAL   CALCULUS. 

6.  dy  =^2  cosec  {nx) .  cot  {nx) .  dx, 

y  z=z-  j  cosec  («a;)  cot  (»a;) .  d  (nx)  =  -^  -  cosec  (»m;)  -f  «• 

Exponential  Forms. 

20.  Prop.  To  integrate  the  form  dy  =  a*(il»^. 

Since       cfot*  =  log  a .  a*(fo?,  .  • .  fa*dx  = flog  a  .  a*dx 

log  a 

EXAMPLES. 

21.  1.  To  integrate  dy  =  3e*(/a;,  where  «  is  the  Naperian  base. 

3e* 

y=Sre'dx  =  , h  c  =  3e*  -f-  c. 

log  e 

2.  dy  =  ba^'dx, 

8.  dy  zn  me^'dx. 


y  =z  —  /  e^'dinx)  =  — e»*  +  c. 


The  cases   which    ha\re  now   been  considered    include  all    the 
elementary  forms. 


CHAPTER   in 


RATIONAL    FRACTIONS. 


22.  Having  disposed  of  the  simple  and  elementary  forms,  d 
such  as  admit  of  being  brought  to  such  by  some  veiy  obvious 
process,  we  shall  proceed  to  the  consideration  of  more  complicated 
expressions,  endeavoring  in  each  case  to  resolve  them  by  a  sys- 
tematic process  into  one  or  more  of  the  elementary  forms. 

23.  The  first  form,  in  point  of  simplicity,  which  we  shall  hav«  ■ 
occasion  to  consider,  is  that  of  a  rational  algebraic  fraction,  and  ii 
such  expressions  we  may  always  regard  the  highest  exponent  of  tht. 
variable  in  the  numerator  as  less  than  the  corresponding  exponent  ii 
the  denominator,  since  the  fraction,  when  not  given  originally  in  that 
form,  may  be  reduced  by  actual  division,  to  a  series  of  monomial 
terms  and  a  fraction  of  the  desired  form. 

24.  Prop.  To  integrate  the  form 

5a;«-i  4.  cx^-2  .  .  .  .  _|_  /a;  -f  I; 
dv  = dx, 

\st  Case.  When  the  denominator  of  the  proposed  fraction  can  be 
resolved  into  real  and  unequal  factors  of  the  first  degree. 
To  illustrate  this  case,  take  the  example 

-         ax  ■\-  c   .  ax  •{•  c     . 

x^  -{-  bx  x{x  -f  h) 

Assume  z=  — | -—  where  A  and  £  are  unknown 

x^  -{-  ox       X       X  -\-  b 


262  INTEGRAL    CALCULUS. 

constants  whose  values  are  to  be  determined  by  the  condition  that 
this  assumed  equality  shall  be  verified. 

Reducing  the  terms  of  the  second  member  to  a  common  denomi 
nator,  we  have 

ax  +  c  _  A{x  4-  h)  Bx      _  Ax -^  Ah -^  Bx 

x^  4-  bx~    x"^  -\-  bx        x^  -{-  bx~  .       x^  -\-  bx 

Hence  ax  -{-  c  =  Ax  -\-  Ah  -\-  Bx\ 

and  since  this  condition  is  to  be  fulfilled  without  reference  to  the 
value  of  X,  the  principle  of  indeterminate  coefficients  will  furnish 
the  separate  equations 

c  =  Ab,     and     a  =  A  -\-  B. 

Thus  we  shall  have  two  equations  with  which  to  determine  the  values 
of  the  two  constants  A  and  B.     Resolving  them,  we  find 

A       ^  jr>  4  c        ab  —  c 

A  =  T-     and     B  =.  a  —  A  =za  —  -  =  — - — • 

0  0  0 

Hence  by  substitution 

dx_ 

x+b 


^      J  x^  -\-bx  J   X  J  X  +  b  bJ    X  b    J  i 


=  ^  log  a:  +  ^^-y-^log  {x  +  b)  +  C. 

As  a  second  illustration  take  the  following  example 

dy  =    „   ■    ,    dx. 
x^  -\-  bx 

A                                         a            A    ^       B 
Assume  = f- 


x^  -\-  bx       X       X  -^  b 

111  ^       _  ^{^  +  ^)    ,      -gt?      _  Ax  +  Ab  +  Bx 

x^-^bx"^    a;2  +  bx       x^  -fJx  ~         x"^  -\- bx       ' 

.' .  a  =  Ax  -\-  Ah  +  Bx,  and  consequently  by  the  principle  of  inde. 
terminate  coefficients 

a=:  Ah    and    0  =  A -[-  B,    whence    A  =^  and  B:='-A=—t' 

0  h 


RATIONAL   FRACTIONS.  263 

And  by  substitution 


/adx        r     adx       _a   rdx       a   P 
~bx  "J  bi^T)  "bJ  ~x"~bJ 


dx 


(x  ^b)       bJ    X       bJ  x-^b 
=  ^  logo;  -  ^log  (a;  +  6)  +  log  c 

=  log  (x^)  —  log  [{x  +  by]  +  log  c 

T.     m    .                  7         (2  +  Sx  —  4:x^)dx 
Ex.   io  integrate   dy  = — 

Here  the  factors  of  the  denominator  are  or,  2  -j-  ar,  and  2  —  «,  and 
we  therefore  assume 

2  +  3a;  -  4a;2      A   ,      B  O 


Ax  —  x^  X      2  -i-  a;      2  —  a? 

_4:A- Ax^  +  2Bx-Bx'^-{-'ZCx-\-  Cx^ 

4x  —  x^ 

,',2-h^x  —  4:x^  =  4:A  —Ax^  -\-  2Bx  —  Bx^  -^  2Cx  -^  Cx% 
and  by  comparing  the  coefficients  of  the  like  powers  of  x,  we  have 

2=4:A,     S=2B  +  2C,      -4.=  -A-B+  C. 
These  conditions  give 

^=1,    ^-|-C  =  |    and     B^0  =  4-A  =  'L 
r,A  =  ^     B  =  l.     C=-l, 


^      2J    x^  2J  2-\-x^J  2- 


—  dx 

X 


=  iloga:  +  -log  (2  +  ar)  +  log  (2  -  a;)  +  c. 

25,  A  similar  decomposition  into  partial  fractions,  each  integrable  , 
by  the  logarithmic  form,  will  be  possible  whenever  the  denominator 


264  INTEGRAL   CALCULUS. 

can  be  resolved  into  simple  and  unequal  factors.  For  if  the  num- 
ber of  such  factors  be  n,  each  constant  numerator,  as  A,  B,  C,  &c., 
will  be  multiplied  (in  the  reduction  to  a  common  denominator)  by 
all  the  denominators  except  its  own  ;  and  since  each  denominator 
contains  only  the  first  power  of  the  variable  x,  it  follows  that  there 
will  appear  in  the  numerator  of  the  sum  of  the  reduced  fractions 
every  power  of  x  to  the  (n  —  \)th  power  inclusive,  and  also  an  ab- 
solute term.  Hence  the  number  of  equations  formed  by  placing 
the  absolute  terms,  and  the  coefficients  of  the  like  powers  of  x  equal 
to  each  other,  will  be  n,  and  therefore  just  sufficient  to  determine 
the  n  constants  A,  B,  C,  &c. 

26.  When  the  factors  of  the  denominator  are  not  immediately 
apparent,  we  may  place  the  denominator  equal  to  zero,  determine 
the  roots  x^,  ot^,  &c.,  of  the  equation  so  formed,  if  practicable,  and 
employ  the  factors  x  —  x^^  x  —  X2,  &;c. 

(4  +  7x)dx 

Put           2a:2  _  4a;  _  10  =  0     or^    x^  —  2x  —  5  =  0. 
Then  x  =  I  dr-y/tT,  and  the  factors  of  the  denominator  are 
x  —  I  +  ^        and         X  —  \  —  y^ 
_  1  r{\^lx)dx  ^  \_  r (4  +  '^x)dx     

• '  •  ^  ~  2^  0:2  -  2.1:  -  5  ~  2^  (^  _  1  +  ^^^  _  1  _  ^ 

"^zJ  x-\  +-v^      2^a;-l-v^* 
r.^-^lxzziAx  —  A--Ay^+Bx  —  B  +  ^y^ 
whence    4.  z=:  -- A  -  Ay/Q  ^  B  +  B^    and    7  =  A  +  B, 
from  which  we  deduce 

A  =  '^-''    and    5  =  lvi±il. 

2v/6  2JQ 


y  = 


BATIONAL   FRACTIONS.  266 

7y/0"-ll  ^ dx^ 7-v/6"+  11  p  dx 

4y^     J  X  — 1+^6  4y/6      J  x  —  1  — y^ 

=^^^^^^iog(.-i+yoy+ ^^^1^^ 

27.  2c?  Clase.  When  the  denominator  of  the  proposed  fraction 
contains  equal  factors  of  the  first  degree. 

To  illustrate,  take  the  example    dy  =  — ^— - —  dx. 

^         ^  (a;  +  hy 

If  we  attempt,  as  in  the  first  case,  to  resolve  this  into  three  frac- 
tions having  denominators  of  the  first  degree,  by  assuming 

a-\-hx-\-cx^  _     A  B  C 

{x  +  hy     ~  X  -{-h      X  -i-h  "^  a;  +  a' 

there  will  result 

a  -\-  bx  -^  cx^  =  {A  -^  £  +  C)(x  -\-  hf, 

and  .•.a=(^H-^+C)A2,  h={A-{-B-{-C)'Zh,  and  c={A+B+C\ 

whence  ^  =  ^  =  f 

Thus  the  assumed  condition  would  establish  a  necessary  relation 
between  the  constants  a,  6,  c,  and  A,  where  none  such  should  exist, 
those  constants  being  entirely  arbitrary. 

It  is  easily  seen  that  such  a  result  might  have  been  anticipated : 

..  A       ^      B       ^       C  A-\-B-\-  C     , 

tor  smce  — — -  -\ -— ■  -\ -—-  = ,  the  proposed  ex- 

X  -\-  h      x  -\-  h      X  ■\-  h  X  i-  h  ^^ 

(t  "4~  ox  "4"  c3/^ 
pression  — - —  —  can  only  be  reduced  to  this  form  when  the 

numerator  is  divisible  by  (x  -f  h)^.  Hence  the  decomposition  of 
the  proposed  expression  into  three  fractions  of  this  form  is  not  usu 
ally  possible,  and  when  possible  it  is  not  necessary  because  the  form 
of  the  fraction  can  be  modified  by  reducing  it  to  simpler  terms. 


266  INTEGRAI.  CALCULUS. 

But  if  we  put  x-\-h=z,  we  shall  have  dx=dz,  and  by  substitution 
(g  4-  5a;  -f  cz^)dx  _  [a  +  b{z  —  A)  +  c(g^  —  2zh  +  k^)]dz 


_  fg  -  5A  +  cA^       h-2ch  c     1 

~  L     (a;-f  A)3      "^  (a:+A)2  "^  a;  +  A  J 


dz 

dx. 


Hence  the  proposed  fraction  can  be  resolved  into  three  fractious 
having  the  forms 

A  B  ^         C  ' 

and 


(x  +  hy      (x  4-  hy  x-\-h 

and  since  the  same  reasoning  would  apply  if  the  number  of  equal 
factors  were  greater,  we  may  in  general  assume 

a  -{-  bx  -\-  cx"^ .  . . .  4-  ix"~^  A  ,  B  I 


{x  4-  hy  {x  4-  A)»       {x  4-  hy-^  x -\- K 

the  number  of  such  fractions  being  n. 


EXAMPLES. 

2    3;^ 

28.   1 .  To  integrate      dy  =  -. dx. 

^  {x—  af 

2  —  3a:  A         ,      B 

Assume  7 r-z  =  -7— • rr  4- 


(x  —  ay       (x  —  ay       X  —  a 

2  —  Sx_       A  B(x  —  a)_A-{-Bx  —  Ba 

'''  (x-ay~  (x  -  ay  "^   (x  -  a)2   ~        {x  -  ay 

,'.2  —  Sx  =AA-  Bx^Ba,  whence  2  =  A  —  Ba,  and  —S  =  B. 
.'.  B  =:  —  3     and     ^=24-  Ba  =  2  •—  3a,     and  consequently 

y  =  (2 -3a) /^%  -  3 /-^  =  (2  ^3a)-i- -3  log  (a:-a)4-(^. 

When  the  denominator  contains  both  equal  and  unequal  factors 
of  the  first  degree,  the  two  methods  must  be  combined. 


EATIONAL  FKACTIOKS.  267 

a;2  _  4a;  +  3     , 

Since      a;-  —  6x^  ■}-  9x  =  x{x^  —  6x  -{-  9)  =  x{x  —  Sy     we  assume 

a;2  ~  4a;  +  3  _  A  B  C      _  J(a;-3)^4-^a;+  Cx{x-^) 

x^—Qx'^-\-9x~'  x"^ {x—Zf      {x—Z)~  x'^-Qx^-\-9x 

.  • .  a;2  —  4a;  4-  3  =  ^(a;2  —  62;  +  9)  -\-  Bx  +  C{x^  -  3a;), 

whence    3  =  9^,     -4=— 6^+^-3C,     and     1  =  ^  +  C/.   » 

.•.u4=i»      C=%    and    ^  =  0. 

=  |loga;  4-  ilog(a;  -  3)^  +  ^logc  =  ilog[ca;(a;  -  3)2] 


=  log  [ca;(a;- 3)2]* 


dx 

dy  — 


{x  -  ^Y{x  +  3) 


A-SSUme    7 -r-^T -— r^  =  7 tr-r  H +   .       ,    oX9  + 


(a;-2)2(a;+3)2~  (a;-^2)2  '  a;-2  '  (a;+3)2  '  (a;+3) 
.  • .  1  =  ^(a;+3)2+^(a;-2)  (a;+3)2+  C(a:-2)2+i)(a;-2)2(a;+3), 
or  1  =  ^(a:2  +  6a;  +  9)  +  B{p(?  +  4a;2  —  3a;  —  18) 

+  Cr(a;2  _  4a;  +  4)  +  D{x'^  _  a;2  -  8a;  +  12). 
.•.0  =  ^  +  Z>,   o  =  ^  +  4^4-(7-i>,    0  =  6^-3^-4C-8i>, 
and  1  =  9J  -  18^  4-  4(7  +  12i>. 

These  equations  give,  by  elimination, 

^  =  — ,      ^  = ^,       C  =  — »     and     Z)  =  — • 

25  125  25  125 

•*•   ^  ~  25./  (a;  -  2)2  ""  125«/  ^^"^  25^  (a;  4-3)2  "''125  /  ~-^ 

=  -25(^)  -  li^"^^^  -  ^)  -  25T^^ 


268  INTEGRAL    CALCULUS. 

29.  Cose  Zd.  When  the  simple  factors  of  the  denominator  are 
imaginary. 

These  factors,  which  correspond  to  the  imaginary  roots  of  an 
equation,  enter  in  pairs,  and  are  of  the  forms 


X  ±:  a  -\-  b  y/ —  1,     and     ar  ±  a  —  6  y—  1. 

and  their  product  gives  the  real  quadratic  factor 

x^  ±  lax  +  a2  4-  62  =:  (a;  ±  a)2  +  62 

Hence,  if  there  be  but  one  pair  of  simjfle  imaginary  factors,  or  a 

single  quadratic  factor,  in  the  denominator,  the  corresponding  partial 

Ax  4-  B 

fraction  will  be  of  the  form  -. r^   .    ,^,  in  which  the  numerator 

\x  ±  a)2  -j-  62 

must  consist  of  two  terms,  one  containing  the  first  power  of  x^  and 
the  other  an  absolute  term,  because  the  denominator  now  contains 
the  second  power  of  x ;  and,  therefore,  if  we  introduced  a  constant 
only  into  the  numerator,  we  should  not  provide  for  having  the 
exponent  of  the  highest  power  of  ic,  in  the  numerator,  only  one  less 
than  the  corresponding  power  in  the  denominator. 

But  when  there  are  several  equal  quadratic  factors,  the  denomi- 
nator being  of  the  form 

\{x  ±  a)2  +  62]«, 

the  partial  fractions  will  be  of  the  forms 

Ax^  B  Cx-\-  D  Ex  +  F 


[(a;  ±  a)2  +  62]«       [(a:  ±  a)2 -f  62J«-i  '    (a;  ±  a)2 -f  62 

the  number  of  such  fractions  being  n. 

That  such  a  decomposition  is  possible  in  all  cases,  will  appear 

more  clearly  by  the  following  illustration.     Let  the  proposed  frao- 

tion  be 

cx^  +  ex^  -h  fx^  4-  gx"^  -\-  hx  +  i 
[(a?  ±  a)2  4-  6'2p 

Put  X  ±:  a  =zy,         and         y^  -{-  b^  =  z\ 


RATIONAL  FRACTIONS.  269 

Then  the  fraction  can  be  reduced  successively  to  the  following 
forms 

c{y  q=  af  +  e{y  gp  a)^  +  f{y  gz  a)3  4.  g[y  rp  gf  ^  h{y  ^a)+i 

cy^  +  ^iv^  +f\y^  +  ^1?/^  +  Ky  + «] 


z'> 
_   (cy  +  0(g^  -  6^)^  +  (/ly  +  9r)(z'^  -  b^)  4-  Ky  +  h 

-  ^6 

_    [c{x±a)  +  e,]{z^-2z^^-^M)  +  [f,{x±a)+g,]{^^-b^)+h,(x±a)-^i, 

~'  z^ 

_   {cb^  —fib^  +  h^)x  +  ^*(^2±  «)<^    —f^b\92  ±  a)4-h  ±  V 
_  [(i  ±  a)2  -f  62]3 

(--  2c  6^  4-/i)a;  -  26'-^c  (g^^  a)  +/i(^2  ^  ^)       ca;  +  c{e^  di  a) 
"^  [(a;±a)2  +  62]2  "^    (a:±a)2  +  62' 

which  is  of  the  form 

Ax-\-  B  Cx^  D  Ex-\-F 


[(a;  ±  af  +  62]3  '    [(a;  ±  a)2  +  62]2   '    {x  ±  a)2  +  52 

And  a  similar  decomposition  would  evidently  be  possible,  if  there 
were  n  equal  quadratic  factors  in  the  denominator. 

30.  It  appears  therefore,  that  when  the  denominator  contains 
simple  imaginary  factors,  the  general  form  presented  for  integration, 
will  be 

{Ax  +  B\dx 

dy  =  -z~ —   ^  ,„-,  ,  where  n  may  be  any  integer. 

^        [(x  ±  a)2  +  62]«'  J  J         5 

Put  a;  ±  a  =  2,     then 

_  {Az  ^  Aa  -^  B)dz 
y  =  (s2  +  62)n 

_    r    ^gc^g  P  (B  ^^  Aa)dz ^ 

•  *  •  ^  -  y  (22  +  ^2)«  +  y  (^24. 52)«  -    2(/i-  i)(g2  4-62)«-i' 

(g2+l2p 


/»_^C?g 


270  INTEGRAL    CALCULUS. 

by  making  B  =fAaz=  A^.     Thus  the  proposed  integral  is  found  to 

depend  on  the  more  simple  form,    /     ^  J^     . 

It  will  now  be  shown  that  this  latter  can  be  caused  to  depend  on 
dz 

diminished  by  unity.     Thus  we  have 


/dz 
7— — - ,  in  which  the  exponent  of  the  parenthesis  is 


dz  _  (^2  _{_  ^2)^2  ^  ^2^2  hHz 


(22   +    62)«-J          (^2  4-  62) n            (^2  -j.  62)«     '      (^2  ^  J2)« 
•  '  *     -^  (^2  -f  62)"  -    62'«/    (^^^62)^^1  ~  b^J  (^2  +  i2)« V    > 

1(^2  -I-  62)«-7  ~  (^^qr^Tpi  (22  4.  62)^-' 


/•       2;2(/2;       _  1  /» 

•  y  (.2 .  62\„-2(,z_i)y  ^ 


dz  1 


(22_|_62)n-2(w— 1)^    (22_^62)«-l  2(n-l)     (z^ -{■  b^)*"^' 

which  value,  substituted  in  (1),  reduces  it  to  the  form 

dz 


r      dz       _  I  r       dz  1         f       ^ 

J  Jf^Wyt  -  "^J  (^2  _|.  62)«-i  ~  252(n-  I)-'  (22  + 


■^2)^ 


n-1 


262(7i-   1)(S2  -f  62)'-l 

.      2n-3      r,       c/2 


262(n  _  1)(22  -f  62)«-l     '     262  (?i  _  1)  ,A    (2:2  +  62)—' 


A2\n-i    ^^"   ^®  rendered   dependent  upon 

/dz 
,  ^       7  2\w-2'  ^^•'  ^^  ^^^*  eventually,  the  original  integral  will 

depend  on  the  form     /  - 


dz  1         .  z 

-  tan-i  -. 


22+62  6  b 

31.  We  infer,  therefore,  that  the  integration  of  a  rational  fraction 
can  be  effected  whenever  its  denominator  can  be  resolved  into  simple 
or  quadratic  factors,  and  that  the  integral  will  be  expressed  in  the 
form  of  logarithms,  powers,  or  circular  arcs. 


RATIONAL  FRACTIONS. 


271 


S2.  1. 


GENERAL    EXAMPLES. 

dx 


dy 


x^  —  \ 


Since    {x^  —  1)  =  {x"^  -\- x  +  \)  {x  —  \),  and  x^-\'X+l 

1       1 


=  (.+  l  +  i/=:3)(.  +  ^      2 


3). 


we  assume 


Ax-\-  B 


+ 


x^—  1        x"^  ■\-  X  -{-  \        X  ■—  V 
.'.  1  =iAx^  +  Bx  —  Ax  -  B  -{-  Cx^+  Cx+  a 
whence   0  =  ^+  C,  0  =  B+  C  —  ^and  I  =  O  —  B, 


H-l) 


X^  +  X+  I 


dx       -dx 


x-V 

2 


and  if  we  put  x  -{-  -  =  z^  ov  x^  +  x  -{•  -  =  z",  x 
Z  4 

there  will  result 

r 


0  —  -  and  dx  —  rfg, 

4> 


I  r  dx        1 


(a;  +  2)rf« 


n^     3-|'og(^-l)-J 


(-a-! 


2^(/0 


J 


z^  + 


2dz 


4^2 


+  1 


1  1  1  Sj"  4-  1 

=  -log  {x  -.  1)  -  -log(a:2  -f-  a:  +  1)  -  — -tan-i  ^^i^  + 


vr 


V3- 


2. 


dy  — 


dx 


X  {a  -\-  bx^y 


21^2  INTEGRAL   CALCULUS. 

Assume  ^^^  _^  ^^2)2  -   3.   +  («  _|.  ^^2^2  +  «  +  6a?2  * 

.-.  l=Aa^,  0=2Aab+B-\-Da,  0=Ab^+Db,  0=C-\-JSa,  0=M, 
.'.  A  =  \,  B=--,  (7=0,  i>  =  --^,  ^=0. 
1     Pdx        h    r      xdx  h     r   xdx 


TT  1    rdx        b   C      ^dx  b     r   xa 

Hence,  y  =  -^ /  7 — .   .  oxo 5  /  — r 

d?^    X        a »/  (a  +  oa;2)2       ^24/^^ 


6a;2 


1  1  a;2 


2a(a  +  5a:2)   '    2a2     ^  a  +  6a;2 

Put  a:*  4-  a;^  —  2  =  0,  and  resolve  with  respect  to  a^. 

.-.  a:2=:-l±|=:l,  or  2r2  =  -2, 

.'.'(«*  +  a;2  _  2)  =  (a;2  4.  2)  {x^  -l)  =  {x^  +  2)  {a:  +  1)  («  ~  l)j 
and  we  may  assume, 

+ T  +  -T-r-o-    Then 


x*-^x^  +  2~  x  +  1       x-l        x^  +  2 

x'^=A{x^-z'^-\-2x-2)+B{x^+x'-^2x-\'2)+C(x^-x)-^D{x»-l). 

.'.0  =  A  +  B+C,  l  =  -^  +  j54-A0=:2^-h25--(7, 
0=  —2A-\-2B  -  D. 

.  •'•  ^"~  ""6*/^~+l       6^a-- 1  "^3^0:2  +  2 

=  -  log  (a:  +  1)*+  log  (a:  -  1)*+  ^  tan-i-^  +  c 


RATIONAL  FRACTIONS.  273 

4.  dy  = -'     Since 

\  —  x^ 

l~a;6=(l-a:3)  (l+a:3)=(l-a:)  (l+ar-h^^)  (1+a:)  {X-x-^x^  put 
1  ^       .       i?       .      Car  +  Z)       ,      ^.r  +  i^ 


1— a:6       Xj^x'X-x'X-^x-^x^l—x-^-x^ 

.'.  l  =  A{l-'  x  -\-  x^—  x^+  x^—  x^)-\-B{\'{-x  -^  x^+  x^+  x^  -\-  x^) 
+  (7 (a;  —  x^+x^-x^)-\-D{l  -x  +  a;3-a;*)  +  ^(2;+ir2-a;*— a;^) 
-\-F(l  +  x-x^  -X*). 

.•.l=A  +  B-{-I)-^F,    0=:-A  +  B-{-O-D  +  E-\-F, 
0  =  A-^B-C+F,    0=^A-^B  +  D^F, 
0  =  A+B+C-I)  —  F-F,    0=— A-^B—  C-E, 

(x-{-2)dx       1    /'(a;-2)cfa; 


1,      ,,^    ,       1,      ,,       .  ^   1    r(2.v+l)dx_^l   f      3a 


Sdfa; 


1_  r{2x-\)dx      ]    y- 3d 


(-tf-i 


3c^ 


(-a^i 


=loga-|-ar)*-log(l-2:)*+j^log(l+rr+:r2)-l  log(l-ar  \-x^) 

2dx  2cL 

^    ^    1   1     yf 


3  3 

^\1-^/  ^\l-a:4-W  2V3V  -y/S 


,2x  —  1\ 

4-tan-i =:l  +  c. 

V3    / 

18 


CHAPTER   IV. 

IRRATIONAL     FRACTIONS. 

33.  The  differential  form  next  to  be  considered  is  that  which  is 
still  algebraic,  but  which  involves  irrational  or  surd  quantities.  As 
the  general  mode  of  treating  such  expressions  is  the  same  in  prin- 
ciple, whether  presented  in  the  entire  or  fractional  form,  they  will 
be  considered  in  the  latter,  which  is  of  very  frequent  occurrence, 
and  which  presents  some  difficulties  peculiar  to  itself. 

34.  When  an  irrational  fraction,  which  does  not  belong  to  one  of 
the  known  elementary  forms,  is  presented  for  integration,  we  en- 
deavor to  rationalize  it,  that  is,  to  transform  it  into  a  rational  form 
by  suitable  substitutions.  The  following  are  the  principal  cases 
•n  which  this  is  possible. 

35.  Case  \st.  When  the  fraction  contains  none  but  monomial  terms. 
As  an  example  to  illustrate  this  case,  suppose 

ni  mi 

ay  = ax. 

a-^x*^  -\-  b-^x* 

Put  X  —  2""*"  or  ar  =  s^,  where  I  is  any  common  multiple  of 
the  denominators  w,  wi,  c,  and  e. 

"Then    x^  =  z^\mce  ^     qj.    x^  z=z  g"      where   -i-  is  an  integer  since 
/  is  a  multiple  of  n. 


n 


Similarly     x"^  —  zP^\ritt^     ^.e  _  gcinm«^     and     x^  =  2«i»»»««. 


IRRATIONAL  FRACTIONS.  276 

Also  dx  =  nmce .  z'^^"-^dz. 

Hence  ay  = {nmce .  z^*^^'-^az\ 

which  is  a  fraction  entirely  rational. 

m!x.  To  integrate         ay  =  — j ^  dx. 

Assume  x  =  z^^  :  then  x    =z  z^,  x    =  z^,  x    =  2®,  a;    =  g^, 
and  dx  =z  12z^'^ .  dz. 

206  —  3^2                     24^9  -  36^5  ^ 
.  • .  av  = 122;^^ .  dz  =z dz 

=  (24^8  -  720'  +  216^6  -  64825)  cfe 

+  1908(2*  -  3^3  +  9^2  -  272  +81  -  -^^)  c&. 


y  =  12/(228  _  Qz''  +  182«  -  Mz^)dz 

243 

2  +  3 


+  1908^^2*  -  3^3  +  922  _  272  +  81  -  -^)  c/g 


«7  t:  / 

n  ^  27  T 

+  1908      -25--2'»  +  323-— 22  +  8l2-243]0g(2  +  3)      +« 

+ 1908  Qa;^- |a;*  +  3a;*- ^a:^+81a:^-243 log(a;^+3)] +c. 

36.  2c]?  Case.  When  the  surds  which  enter  the  given  expression 
contain  no  quantity  within  the  (  )  but  one  of  the  form  (a-\-hx). 
As  an  example,  take 

^  {a  +  hxT  +(«  +  tor 
(a  +  ix)^  +  h 


276  INTEGRAL   CALCULUS. 

Put  a  -{-  bx  =  z^""^  where  the  exponent  of  2  is  a  multiple  of  all 
tne  denominators  n,  m,  and  c. 

Ml  mi  e^ 

Then  (a  +  bx)'*  =  s"!'"^  {a  +  bxy  =  2'»i«%  (a  +  62:)'  =  g«i«"», 

_  _         nmc  .  . 

and  ax  =  — —-  .  z^^'^-^dz, 

0 

and   Dy  substitution 

g»iwjc  _L.  ^OTinc 

(/v  =  7- ; — TV  •  nmc .  z^^'^-'^dz. 


vhich  form  is  entirely  rational. 

x^dx 
37.  1.  To  integrate      dy  = 


(1  +  4:.)* 


Assume  1  -\-  4x  =  z"^.     Then 

r  =  ^^^,  ^^  =  ^5  x^  =  ^{z^-Sz^-\-Sz^-l),  md  {l+4x)^=z^. 

..,d,==±l'^^:^'-±^J^^]zdz  =  ±(z^-s  +  ^-l-U 

^  128\  Z^  I  12«\  ^2  ;2*y''^' 


^^^^      ^  (1+4^)*     3(l+4:r)^J 


2.  c?y  = 


Put  1  +  a:  =  z\    Then  a:  =  22  _  1^  f/a-  =  22^0,  and  -/l  +  a?  =  1 

_         22c/2?         _      2(/2       _       C?2  </2 

.  • .  y  =  log  (2  —  1)  -  log  (0  4-  1)  -f  c, 


.      2-1,  ,      Vl-ha:-l 

y  =  log— — -r   +  C  =  log  ^-—=-_: +  C. 


IRRATIONAL  FRACTIONS.  277 

38.   CoLse  Zd.  When  the  proposed  fraction  contains  no  surd  except 
one  of  the  form 


y/oT-^-  bx  ±  c^x^  =  c  \/— -  +—-x±.x\ 
When  the  last  term  is  positive,  assume 


\/^  +  -^^  +  ^''  =  ^  +  ^'7     then   ^^—x-\-x^=zz^-{-2zx-{-x» 


a  —  c^z^ 

X  — 


2c^(a  -bz-\-  c^z^)  ^ 


2c^z—b  i^c^z  —  by 

and      ^bx^^cH^=c(z-^x)=ci^z-^——j=        ^,^  _  ^— ^- 


The  values  of  x,  ya  -{-  bx  -\-  c^x^,  and  dx,  being  all  expressed  ra- 
tionally in  terms  of  z,  the  proposed  differential  when  transformed 
will  also  be  rational. 

Again,  if  the  term  involving  x^  in  the  surd  be  negative,  denotw 
by  x^  and  arg  the  roots  of  the  equation 

"^     ;^ ~:^  =  ^'  ^^^"^  ^' - -^ - ^  =  (^ - ^iK^ - ^2), 

and  therefore      —  H — -  —  x^  =  {x2  —  x)(x  —  Xi), 

Now  assume    -^Jx^—  x)  (x  —  x^)  =  (2;  —  x^)  .  z. 

,'.  X2  —  x  =  {x  •—  Xj)z^,     whence     x  —    ^         ^ 


1   +^2 

''-'-^^^^    and    v'a  +  *^-c»^  =  -(^-^.)='43-^- 

Hence  the  several  expressions  which  enter  into  the  proposed  dil^ 

ferential  will  be  rational,  and  therefore  that  differential  will  become 

entirely  rational. 

dx 
39.  1.  To  integrate      dy  =  ---     . 


278  INTEGRAL   CALCULUS 


Assume  Vl  +  a:  +  x^  =  z  -{-  x;  then  1  -\-  x -\-  x^  =  s^ -i  2zx-\-x^ 
1-^2  2(1 -0+^2)  y,  ,      ■     „      1-0  +  ^2 


/'20-1       2(1-04-22)  /.  2c?0  .     ,^      .,, 

=  log  ^r 7-  +  c  =  log ; h  e 

6  2^  -  1  ^  ^  2VTT^T^  -  (2x  4-  1) 


=  log  [2  /r+V+^  4-  (2a;  4-  1)]  -  log  3  4-  <? 


=  log  [2  y/1  4-  a:  4-  x^  +  2a?  +  1]  4"  Cj. 
2.  c/y  = 


Put  1  4-  a:  —  a;2  =  0,  and  find  the  roots  ar,  and  a?2i  thus 
ajj  =  -  4-  ^  V^     ^^^^    ^2  =  9  ■"  9  V^'      ^ow  assume 


^14-a;— a;2  or  ■^/(a;— arj)  (a;2~a;) 


--_y/5-ar=(a:-----/5)2r2  and 


^i-iv^.Vl=p-' 

(^-2-2V^)^- 

.. ,  ^-^"^-(^-i"-^)- 

""■J    •^-                       1+^2 

T-:^-:?         V^« 

dx  =  77^-r~"2Ti     ^"<^  -/I  4-  a?  —  a;2  =  —  ^  "^ ' ""  -» 

(1  4-  02)2  V  1  4-  ^2 

/'I4-22     2v/5T0cf0        ^  /•  «?2 
*^  s/b.z      (14-2^)^  '^  14-2'^ 


=  ^ 2 tan-  ^^^Y^7''l+^^ -2 tan-i 

^~2-2^^  -'22 


IRRATIONAL    FRACTIONS.  279 

dx 


Assume    \/t^  +  x^=z+x ;  then  —  -\-  x'^  =  z^  •{■  2zx  +  «* 

X  =  -— - — ,  dx= TTTr^TT^ — -  «^5  and  '\/a^-\-b^x^  —  — — • 

26^2    '  (262^)2  5  V       I  262 

r   2bH  2bz        252  (a2  4- 62^2)  /*    26c?g 

•  *  •  ^~ ~y  a2-62^2  X  ^qr^v  ^        (2622)2        ^-      J  a'^b^z^ 

\    r   bdz       ,     \    P  —  bdz        1  ,      a  ^bz  , 

== / 1 / = lOff 1-  c, 

aJa-\-bz        a  J  a  —  bz       a      ^  a  ■\-  bz 


1  ,      a4-6a;— Va2_|.52^2  i         -i/a2  +  62a;2  —  a   , 

=  —  log ^^ +  c  =  — log  -5^ h  c. 

a      "^  a-bx-ir-\/d?'-^h''x^  «  ^^ 

40.  The  other  irrational  forms  which  admit  of  being  rationalized, 
are  chiefly  those  belonging  to  the  binomial  class,  which  it  is  proposed 
to  consider  carefully  in  the  next  chapter. 


CHAPTER   V. 


BINOMIAL    DIFFERENTIALS. 


41.  Pro-p,  To  determine  the  conditions  under  which  the  general  form 

dy  =  x^  {a  -\-  bx^)^dx,  can  be  rendered  rational. 

If  we  put  aj^rs™",  there  will  result  x^  —  0«i»,  x^  =  z^t^    and 
dx  =  mn  .  z'^^-'^dz. 

.  • .  dy  z=  z^i^{a  4-  hz*^'^Y  nmz*'^^dz, 
so  that  the  form  will  be  equally  general  if  written  thus 

dy  =z  x^{a  +  bx^)Pdx (1), 

in  which  p  is  the  only  fractional  exponent. 

Assume      a-\-bx^  =  z'y  then  x  =  I — -—  j 


and  by  substitution  in  (1), 

1  CT+i 

dy^—^iz-a)""      'zW 

nb  » 
Hence,  if be  a  positive  or  negative  integer,  or  zero,  the 

m+l     ^ 
quantity  (z — a)  "        can  be  developed  in  the  form  of  a  series  of  mo- 
mials   (with   a   limited   number  of  terms),  a  rational   fraction,  or 


/ 


& 


/ 


BINOMIAL  DIFFERENTIALS. 


jiiity,  and  thus  the  value  of  di/  can  be  rendered  entirely  r 


/ 


\'S 


be  Qj 


For,  although  p  is  a.  fractional  exponent,  the  expression  can  be  *o^/, 
transformed  as  to  remove  the  fraction,  by  the  method  explained  in  • 
the  first  case  of  irrational  fractions. 

Again,  since  a:»'(a  +  bx^)P=  x"^+^P(ax-^  +  6)?,  if  we  put 
ax-^  -}-  b  =  z,  there  will  result, 


'? 


/ 


=f-^T''-^"--f-^') 


m+np+l 


m+np+l 


m+l 


.  • ,  x"^'^^Pdx 


.  dy—  — 


a 


n 

m+np+\ 


-(.-6) 


-p-i 


m+l 


{z-h) 


p-\ 


dz. 


zP.dz, 


m  ~\~  1 

And  this  can  be  readily  rationalized    when hi?  is  a  positive 

or  negative  integer  or  zero. 

We  conclude,  therefore,  that  there  are  two  cases  in  which  it  will 
be  possible  to  rationalize  the  general  binomial  differential,  viz. : 

1st.  When  the  exponent  m  of  a;  exterior  to  the  parenthesis, 
increased  by  unity,  is  exactly  divisible  by  the  exponent  n  of  a: 
within  the  (  ) ;  or 

2d.  When  the  fraction  thus  formed,  increased  by  the  exponent  p 
of  the  (  )  is  an  integer  or  zero. 

42.  These  two  relations  are  called  the  conditions  of  integrahility  o{ 
binomial  differentials. 

43.  1.  To  integrate     dy  =  x^{a  +  x'^ydx, 
m  +  l 


Here  m  =  5,  «  =  2, 


=  3,  an  integer, 


and  the  expression  can  be  rendered  rational. 

Put         a-\-x'^z=z,  .'.  xz={z-  ay,  x^  =  {z  -  af 
x^dxz=^{z-afdz,  and  dy=hz-af  .  Xiz  =  \{z^ -^2az^-\a^^)dz 


282  INTEGRAL  CALCULUS. 

Here  m  =  —  2,     ?i  =  2,     and     ^  =  —  -• 

.  • .   =  —  -,  a  fraction ; 

n  2  ' 

,    .  wi  +  l  13 

but \-p  =1  —-  —  -=:  —  2,  a  negative  integer, 

and  the  expression  can  therefore  be  rendered  rational  by  the  second 
transformation. 

Put  a:-2+l=:^,     ,'.  x={z  —  \)^,     r^=.{z^\Y 

1  1-1 

ar^  .dx  =z  —  ~{z  —  V)dz^     and     dy  =z  —  -(z  —  l)z     dz. 

,  1   -i  -        1    -f . 

or,  dy  =z  —  -z      dz  A-  ~z     dz. 

z  z 

.'.   y=-'^fz-hz  +  lfz'-*dz=-(z^  +  z-^)-^c. 

z-\-l   ,  ar-2  +  2    .  1       /I      ^  \  . 


« 


CHAPTER   VI. 


FORMULA   OF   REDUCTION. 


44.  When  a  binomial  differential  satisfies  either  of  the  conditions 
of  integrability,  it  is  possible  to  transform  it  into  a  rational  expres- 
sion ;  but,  instead  of  applying  this  process  of  rationalization  directly, 
it  is  often  more  convenient  to  employ  certain  formulce  of  reduction^ 
which  render  the  proposed  integral  dependent  upon  others  of  simpler 
form,  or  such  as  have  been  previously  integrated. 

45.  Such  formulae  are  deduced  by  employing  another  known  as 
the  formula  for  integration  by  parts. 

Thus,  since  d(uv)  =  udv  +  vdu,  we  have 

fud(o  =  uv  —  fvdu (1). 

By  this  formula,  fudv^  is  ma5e  to  depend  upon  fvdu,  which 
latter  integral  may  be  more  simple. 

46.  Prop.  To  obtain  a  formula  for  diminishing  the  exponent  m 
of  ar,  exterior  to  the  (  ),  in  the  general  binomial  form 

y  =  fx^(a  +  bx^)Pdx. 

Put  (a  +  bx'*)Px^-^dx  =  dv,     and     a;'"-"+^  =  u. 

Then  v  =  ^— - — ---.-,     and     du  z=  (m  ^  n  -\-  l)x^'*-^dx, 
7ib(p-\-  1) 

But     y  z=z  /a;^-«+i(a  +  bx^)Px^~'^dx  =  fudv  =z  uv  —  fvdu. 

x-»+^,^M^-)^'  _  '!^jzJL±\f^u  +  4;r«)^><&  (2). 


284  INTEGRAL    CALCULUS. 

The  formula  (2),  effects  the  object  of  diminishing  the  exponent  m 
of  ic,  but  it  increases  by  unity  the  exponent  p  of  the  (  ),  and  as  this 
would  often  be  an  inconvenience,  we  must  endeavor  to  modify  (2). 

Now  /a;"»-«(a  +  bx*)f^^dx  =  /a;'"-"  (a  +  bx'')P{a  +  bx^)dx 

=  a/a;'"-'»(a  +  bx^)Pdx-^b  Jx'^ia  -\-  bx^)vdx. 

.  •.    y  =  fx'^ia  +  bx'')Pdx  = -y-^ -^ 

-  (^-  -  ^^  +  |)^y-^.-n(,,  +  b^n)pdx  -  ^^-±lfxrn(a  +  bx-)Pdx. 

Hb(/j  -Hi).'  7i{p  -f  1)  -^      '  ^ 

Transposing  the  last  term  to  the  first  member  and  reducing,  we 
have 

-^ ^/a;'"(a  +  bx'^\Pdx  = ~-^T^ — 

Hence     y  =  fx^(a  +  bx")Pdx 

_  x"'-^^\a  +  6x'»)^i  —  (m  —  n-\-  l)a.  fx"^^{a  +  bx«}Pdx  . 

_  fc(,,^  -f  m  +  1)  ■  ^    ^' 

47.  By  this  formula,  the  proposed  integral  is  made  to  depend 
upon  another  of  a  similar  form,  but  having  the  exponent  m  —  w  of 
X,  exterior  to  the  (  ),  less  than  the  oiiginal  exponent  m,  by  w,  the 
exponent  <>f  x  within  the  (  ). 

48.  Prop,  To  obtain  a  formula  for  diminishing  the  exponent  p 
of  the  (  ),  in  the  general  integral 

y  =  fx^ia  -}-  bx^)Pdx. 

Since     fx^{a  -f-  bx^)Pdx  =  fx^(a  +  bx'^)P-\a  -}-  bx^)dx 

=  afx'^(a  4-  bx'')P-^dx  -\-  b/x'^'^^a  +  bx'')P-'^dx\ 

and  since  by  applying  formula  {A)  to  the  last  integral,  replacing 
m  by  (w-|-«),  and  p  by  (/>  — 1),  we  get 

*  ^  b(np  +  m  4-  1) 


FORMULiE  OF  REDUCTION.  285 

.  • .     y  =z  /x'"{a  +  bx'')Pdx  =z  a  fx'^{a  -f  bx'')P-^dx  H -"- -—- 

Tip  ~j~  TUf  ~T~  X 

np  +  m  +  l''      ^  ' 

x^'*-^\a  •\- hx^^P  ■\- pnoi  J x^{(x -\- hx'^y~^dx  ,„. 

~  np  -\-  m  -\-\ 

49.  By  the  use  of  this  formula,  the  proposed  integral  is  made  to 
depend  upon  a  similar  integral,  but  having  the  exponent  of  the  (  ) 
diminished  by  unity. 

50.  When  the  exponents  m  and  p  are  negative,  and  numerically 
large,  it  is  generally  convenient  to  increase  them,  so  as  to  bring 
their  values  nearer  to  zero,  and  hence  we  require  two  additional 
formulae,  one  for  increasing  the  exponent  of  the  variable  exterior  to 
the  (  ),  and  the  other,  for  increasing  the  exponent  of  the  (  ). 

51.  Prop.  To  obtain  a  formula  for  increasing  the  exponent  — -  m, 
of  the  parenthesis  in  the  general  integral 

y  =  fxr^{a  +  bx")Pdx. 
From  formula  (A),  we  obtain,  by  transposition  and  reduction, 

xrn-n+i(a.uix'*)P+^—b(np-^m+\)fxP(a-{-bx'')Pdx 
fx'^-Ha-^bx'^)Pdx=z i— !- — -I —^^       ,    ' /-^ — ^^-^ ^- 

Now  making  m  —  n  z=  —  m^,  or  m  =^  n  —  m^,  there  results 
fxr^i{a  +  bx^)Pdx 
_  xr^v^\a  4-  bx'^)P^'^  —  b{np  -\-  n  —  m^  -\-  1) /ir-*"i+» (cr.  +  bx^)Pdx 
""  a(-mi-f  1)  ' 

or  by  omitting  the  subscript  accents  and  reducing, 

y  —  fxr^ia  +  bx^)Pdx 

y.-m+i  ((J  _|_  5^ft)  p+i^h(^^_np—n— l)fx-^+''(a-\-bx'*)Pdx  , 

-  -  —  -a(m~l)  ^^>- 

52.  By  the  use  of  this  formula  the  exponent  —  m  of  x  exterior 
to  the  (  )  is  increased  by  n  the  exponent  of  x  within  the  (  ). 


286  INTEGRAL    CALCULUS. 

53.  Prop,  To  obtain  a  formula  for  increasing  the  exponent  —  p 
of  the  (  )  in  the  general  integral 

y  —  fx'^{a  -f  bx^)-Pdx. 

From  formula  (B)  we  obtain,  by.  transposition  and  reduction, 

fx'^{a-i-bx^)p-^dx  = ^— ^ ^^-^-^ — —^- — ^^-^ ^ 

Now  making  p  —  I  =  —  p^  or  p  =z  i  —  p^^  there  results 

/a;'"(a  -|-  bx*)-Pidx 

_  a;'"+i(a  +  6a;«)-Pi+i  +  {np^  —  n  —  m  —  \)fx'^{a  +  bx'^yPi+^dx 
■"■  -.na{^p^+l)  [     . 

or  by  omitting  accents  and  reducing 

y  =  fx^{a  +  bx^)-Pdx 

__  a;"*+i (a+ Sa:«)-i^i  +  {np—n  —  m  —  l)fx'^[a-\-bx'')-P^'^dx 

~  na{p  -  1)  ^^' 

64.  By  the  use  of  this  formula,  the  exponent  — j?  of  the  (  )  is 
increased  by  unity. 

Applications  of  Formuloe  (A),  (B),  (C),  and  (D). 

x'''^dx 

55,  1.  To  integrate  dy  = where  m  is  an  odd  integer. 

y  1  —  x^ 

Put  771  successively  equal  to  1,  3,  5,  7,  dsc,  and  apply  f  /,.     Thus 

y'    xdx  J 

-===i  —  —  y  1  —  x^  -\-  Ci  by  the  rule  for  powers, 
v/l  —x^ 

which  m  n=  3,  /i  -=  2,  and  p  z=z  —-^ 


APPLICATIONS  OF   FORMULA.  287 

y^_f^=_J,4  0:^^  +  *y^^    by  formula    (A),   in 

which  m  =  5,  7i  =  2,  and  p  =  —  ^' 
Hence  by  substitution, 

Jy/T^r^^~    yi       5-7    +3.5.7* +  1.3.5.7;^-^  ^^" 

and  generally 

r  ^"^^   _ _  fl^^.  I  l-('»-l)^„-.,  I  l.(>»-3)(m-l)^_,  ^  ^,^ 
•/t/l  —  a;2  Lm  (m—2).m  (m—4){m—2)m 

••••  +  1.3.5.7....{m-2)(m)        J  ^^       ^   +  ^- 
2.  To  integrate  <iy  =  ,  where  771  is  an  even  integer. 

Put  m  =  0,  2,  4,  6,  Ace,  and  apply  (^4)  thus 

r  =  sm-^x  +  Cq  by  one  of  the  circular  forms. 

«/-v/l  —  x^ 

which  m  =  2,  n  =  2^  and  ^  =  —  -. 


288  INTEGRAL   CALCULUS. 

which  m  =  4,  ri  =  2,  and  ^  =  —  -. 

f  J =  —  ^3:5/1  —  a;2  +  -  /- .^    "^   ;     and  generally 

/»  a;'"d'a;  1  ,    /:; ,  m  —  1   /» x^-'^dx 

Hence  by  substitution, 
/»    a;2c?a;  1        , 1 

/.  x*dz  n    3  ,1.3  \    /^ r  ,  1.3  .     ,     ,    _ 

f;/TT7.  =  -  (i  ^' + 2:4^)v^^^+ o  ^'"-'^  +  ^*- 

Vr^^^"       ^6''  +4.6 "^ +2.4.67^  +2A(}""    *+-" 

and  generally 

/»   a-'«c/^  /I        ,  .  l.(w-l)       ,  .  l.(m-3)(m-l)        ^   .    „ 

/^i  _  -J.2  \m  {m—2)m  [fn~4:){m—'2)m 

_  _  _  1.3.5.7...(m-3)(m-lU    ,p-^ 

^2.4.6.8.  ..  (M-3)»t  7^ 

1.3.5.7...(m-3)(m-l).     ,     . 
+  3.4.6.8...(m-2).m  ""    ''^^- 

3.  c?y  =  — ~=  =  7r\\  +  X)   ^dx, 

a;2y  1  -f  x 

Make  m  =  +  2,  ^  =  —  -,  and  n  =  1,  and  apply  (C7)  :  then 

tit 


—  _  -/'  +  ^  _  1  y ^f 


APPLICATIONS   OF   FORMULJii.  289 


Now  put  1  -f  a;  =  z^^  then  x  =  z^  —  \^  dx  —  2zdz  and  -y^l  +  x=zz. 

P       dx r      2zdz      _  r  2dz     _  /*   dz     _  /*   dz 

'J  TjT^x  -J  {f-\)z-J  z'^-l~J  z-\      J  z-\-l 


2  —  1                       "v/l  4-  a;  —  1        ^ 
=  log  — r^  +  (7  =  log  -^, h  C7. 


^  ^         y  1  +  a:  +  1 

I 
"4.  dy  = ~  dx  =  x-\a  +  bx)  dx. 

Put    m  =  —  1,     n  =  1,     and    P  =  ^,     and  apply  (5)  ;  then 

/»  (g  -h  bxfdx       x^a-^bxy   ,2^,     ..    ^    .4 

2"^  ^  ""^        2 

Now  put    m  =  —  1,     w  =  1,     and    jo  =  ^,     and  apply  (i!5)  to 
the  last  integral ;  thus 

i      -a 

f^'(a  +  bx)hx  =  ^'^^  '^  ^"^^    +  \/x-^(a  +  6x)'"*c^:r. 

2-^^-1        2 

Now  put    a  +  ^^  =  2^;     then      x  =  — r — ,     dx  =  — r-,     and 

0  0 


y'a  -f-  6i«J  =  2. 

1  ^  1 

=  — :log >-  +  (7  =  -^  log  ^^ >^     -f  a 

ya        2  -f-  y a  y a        y a  +  6;c  -f  y a 

and  by  substitution, 

19 


290  INTEGRAL    CALCULUS. 


^  y  a  4-  ft^  +  V  a 

5.  cfy  = ^  =  ar-i(l  +  2a:)  Va:. 

Put     w  =  —  1,     71  =  1,    i>  =  ^,     and  apply  (i)),  then 

(* 

3 

.(1  +  2.)*       1.(5-,)  i 

But  Sx-^{\  +  2a:)"'*c?a:  =  log  \LJl^  —  +  (7,  by  the  laat  ex- 

yTT^a;  +  1 
ample. 

2  .  ,      Vl  +  2^  -  1    ,    n 

yT+2a;  vTT2x-|-l 


CHAPTER   VII. 

LOGARITHMIC    AND    EXPONENTIAL    FUNCTIONS. 

56.  We  shall  now  proceed  to  the  integration  of  those  forms 
which  involve  transcendental  functions,  beginning  with  the  case  of 
logarithmic  functions. 

57.  Of  the  logarithmic  forms,  only  a  very  limited  number  can  be 
integrated,  except  by  methods  of  approximation.  The  principal 
integrable  forms  will  be  examined. 

58.  Prop.  To  integrate  the  form  dy  =z  X .  log*»a:.  dx,  in  which  JST 
is  a  given  algebraic  function  of  x. 

Put  Xdx  =  dv,  and  log"a?  =  w,  and  apply  the  formula  for  inte 

gration  by  parts.     Thus 

dx 
V  =  fXdx^  and  du  =  n  ,  log^-^a: .  — ,  and  since  ftcdv  -zzUv  —  Jvdu^ 

X 

.  • .  fX,  log«a: .  dx  =  log«a; .  fXdx  —f\n.  log^-^a; .  /(Xdx) .  —  I 
or,  by  making  J  Xdx  =  X^ 


— ^ .  log"-^a: 


dx. 

X 

If,  therefore,  it  be  possible  to  integrate  the  algebraic  form  Xdx, 
the  proposed  integral  will  depend  upon  another  of  the  same  general 
form,  but  having  the  exponent  of  the  logarithm  less  by  unity. 

rx 

Now  put   /  — ^dx  =  X2,  and  by  a  similar  process,  there  will 

result 

/X  P  X 

—^  log"  -^x.dx  =  Xz  log"  -^z  —  {n  —  \)  J  — ^  log»-2a;.  dx. 


292  INTEGRAL  CALCULUS. 

If  n  be  a  positive  integer,  the  repeated  application  of  this  formula 
will  cause  the  proposed  integral  to  depend  ultimately  upon  the  alge- 

/X  XX 

— -  dx,  provided  we  can  integrate  Xdx,  — ^  dx^  — -  c/ar, 
X  «C  2t 

&c.,  obtaining  in  each  an  integral  in  the  algebraic  form. 

log  X .  dx 


Ex.  To  integrate  dy  = 


(1  +  xy 


Here  X  =  ^-^,  =  (1  +  .)- 

.  •.  /Xc/ar  = /(I  +  ar)-2rfa:  =  -  j-i-  =  Xj. 

*  1  ,  -.  ^^^  X    ^     P     dx 

Also,     w  =  1,     and     .  * .    y  z=.  —  ■—■ \-   I  — ; — :• 

'  '  ^  \+x^  J  x-^x^ 

But  /^-f^,=/5-/3^^  =  log.-log(l+.)-f  6'. 

59.  Prop.  To  integrate  the  form     dy  =z  x"^ .  log"a; .  dx^     in  which 
n  IS  a  positive  integer. 

Put     x^  =  X;    then     X.  =  f  Xdx  =  f  x^dz  = -^^^ - 

m  +  l 

And,  therefore,  by  the  last  proposition, 

y  =  /af» .  log"a;.  dx  =  — -—  log"a; ■— ■/  x^ .  log'-^a; .  dx : 

m-\-l     °  m-\-\  ° 

and  similarly, 

x^^^  n  —  1 

fx^losi^-'^xdx  = lo2'»-^ar f  x^los^-'^xdx. 

^  m  +  1^  m+1''         ^ 

/*a;'"log'»-2a:(/a;  =  — — -  lofi;'*"^^; __  f  3f^\oa*-'^x.dx, 

m+l°  w-fl  ° 

dzc.  &;c.  &c. 

Hence  by  successive  substitutions, 


LOGARITHMIC  FUNCTIONS.  298. 

/j;ff«+i  n  ^  n{n  —  1) 

'  *^  m+lL  m+1     °  (wH-1)^ 

n(?i-l)(?i-2)(yi-3) 3.2.1"] 

Cor.  This  formula  ceases  to  be  applicable  when  m  =  —  1,  as  the 
terms  become  infinite  ;  but  we  then  have 

dx  loff^^^iC 

/ orHog'^x .  dx  =zf  log"x -—  z=f  log«a; .  c?(log a;)  =     ^         +  (7. 

Ux.  To  integrate         dy  —  x^  .  \og^x .  dx. 

Here     m  =  3,     n  =  S,     ^+1=4,     r*  — 1=2,     n  — 2=1. 

rr*  r         ^  s  2         3  2  n 

.  •.  y=>3.1og3a;.rf^=— [^log3i;--log2a;+-^loga: ^J  +  0. 

2.  dy  =     "     •  da;  =  a:     log^a; .  cfe. 

X 

3 

Here  m  =  —  -     and     n  =  6, 

.•.wi4-l  =  —  -,  w  —  1=4,  »  —  2  =  3,  w  —  3  =  2,  w  —  4  =  1. 

2 

.  • .  y  = ^  [log^a;  -f  5 .  2  log*a;  -f  5  •  4 .  221og3a; 

a:* 
-t-  5 . 4 . 3 .  23Iog2a;  +  5.4.3.2. 241oga;  +  5 .  4 . 3 .  2 . 1 .  2^]  +  C. 

Remark.   If  we  suppose  n  to  be  a  positive  fraction,  the  same 
formula  will  apply,  but  the  series  will  not  terminate. 

x^dx 
60.  Prop.  To  integrate  the  form  dy  =  - — —.  in  which  w  is  a  posj. 

tive  integer. 

Put  x^+'^  =  u    and • —  =  dv.  then 

log"a;     x 

duz=(m-\-l  )x^dx.    and    v  = — -; — . 

^  ^        '  (/I  — l)log«-iar 


294  INTEGRAL    CALCULUS. 

Applying  the  formula  fudv  =  uv  —  fvdu,  we  obtain 

log"a:  ~       {n  —  l)log"-ia;       n  —  \J  log''*~V  ^ 

log«-"i^  ~  ~  («— 2)log«-2a;  "^  n  —  '^J  log"-2^* 

log«-2a;  ~~       {n  —  3)  log«-%      n  —  sJ  log^-^a;*     ^*    ^* 

/a;'»c?a;             x'^+^  f      1         .   wi  +  1         1 
:zz  —  • I ■ . 
log»a;           n  —  l  Llog^-^a;       w  —  2    log^-^a; 

■         (m  +  l)^  _1_    . 

"^  (w  -  2)  (w  -  3) '  iog«-3a;'        * 


•  *  '  "^  (w  -  2)  (»  -  3) 


(w-4)...3.2.l'loga?J 

+  („_i)(„._2)(?i-3)  ...3.2.1 ./  iog^* 
The  last  integral  admits  of  only  an  approximate  determination, 
but  its  form  may  be  simplified ;  thus, 

put   z  =  a;"*+^,  then  dz  =:  (m -\-  l)x^dx,  and  (m  4*  1)  ^ogx  =  logg. 

'x^^dx  _    /*  (?2; 
log  X  ~  J  log  5; 

This,  also,  can  only  be  integrated  approximately  by  expanding 
the  expression  under  the  sign  of  integration  into  a  series,  and  then 
integrating  the  terms  separately,  a  method  which  will  be  considered 
more  at  length  in  a  future  chapter. 

m  X^dx 

3.   10  integrate  approximately  dy  =  :j — g— 

Here  m  =  4  and  n  =  3,  .-.^4-1=  5,  w  —  1=2,  n  —  2  =  1. 

Px^dx  _  _  ars  V     1  5       1    1   ,   25  /'  x^dx 

*  ^  ~V  k^  ~  ~  T  Llog2^       T '  ioi^J        2«/ 


log  X  ~  J  \ 


--  c  ,  /*  ^  t/a;  r  dz 

Now  put         .r^  z^  z,  then    /  -; =  /  ,- ; 

•/    log  ar        »/  log  r 


^^        .    logarj        2«/    log*' 
log 


EXPONENTIAL    FUNCTIONS.  296 

and,  making      log  ^  =  ^,  we  have  z  =  e*,  dz  —  e^,  dU 

=log  [log a;5] 4- log a;5 4-^-^2 log^o;*  +  ^  ^  ^^log^ar^-f  &c. 


Exponential  JFuncUons, 

61.  To  integrate  the  form  dy  =  a^ .x^.dxy  when  m  is  a  positive 
integer. 

Put  a''dx=dv.  and  x^=u:  then  t;=T ^a*  and  du=mx^~^dx, 

log  a 

a* .  a:*"         ^      ^  ,  ^  ,     .     .,     , 

.  • .  fa*  .x^,ax  = fa* .  x^-'^dx,  and  similarly 

log  a       log  a 

»       /a* .  x^^^dx  =  -V- ^ fa* .  x^-^dx 

log  a  log  a 

/a* .  x^^'^dx  =  -J ; fa* .  x^^^dx,  &c.  &c.  &a 

log  a         log  a 


^ence,  by  substitution, 


a*  r 

y  =  fa* .x'^.dx  = 1  x^ 

logaL 


mar' 


«v— 1 


log  a  L  log  a 

log^a  log^a  "^ 

log'"a  J 


296r  INTEGRAL   CALCULUS. 


Or 

62.  Prajp.  To  integrate  dy  =  —dx^  when  w  is  a  positive  integer. 

Put  xr^dxz=dv  and  a'=i^ ;  then  v= and  du  =  log  a .  a^ctr. 

m  —  \ 

/w^dx                     a'             ,     lofja     r  a'^dx         ,    .     .,     i 
=r  —  7 -T r  H ^— -  / 7,  and  similarly, 

/a^dx  a'  log  a       Pa'dx 

a;'»-i  ~  ~  (?/i  —  2)^""^       m~— 2  *  J  a^ 

Pa'dx  a*  log  a       fa'dx    »       „       » 

y  ^2  =  "  (;;ri--3ji^  +  ;;^-3  V  i^' ^'- ^'- ^'- 

Hence,  by  substitution, 

/a'dx_  a"  r         log  a  log^a  ^ 

■■^-~(^_l)a;m-iL    +^r^"2^  +  (m-2)  (m-sf  "^     ^ 

3). ..2.1'^      J 


+ 


(m  —  2)  (wi  —  3) 


4- 


-2)...2.W 


(m—  l)(m-2j...2.  W     a; 
The  last  integral  can  only  be  found  approxiiiiately. 

1.  To  integrate  dy  =  a" .  xMx. 

Here  m  =  3,     m— -1=2,     m  —  2  =  1.     Hence 

logaL  log  a      log^a      log^aj 

2.  To  integrate  dy  =.  e'  .x* .  dx. 

Here     m  =  4,     m  —  1=3,    m  —  2  =  2,    m  —  3  =  1,  log  c  =  1. 
.  • .  y  =  e*(a;*  —  4a;3  +  12a:2  __  24x  +  24)  +  C. 

3.  dy  =  e-*a;2fl£r  =  r-*(—  x^dx  =  —  e-'{—  xYd{—  x). 

Here  m  =  2,  m  —  1  =  1,  logc  =  1,  and  x  in  the  general  formula 
is  to  be  replaced  by  —  x, 

r.y=fe-*.x^,dxz=z  -  r-*(a;2  +  2ar  +  2)  +  C. 


EXPONENTIAL  FUNCTIONS.  297 


4.  dy  =:  ~—  dx. 

X* 


Here  w  =  4,     7n  —  1=3,     &c. 


the  last  integral  being  found,  approximately,  as  in  a  previous  exam, 
pie,  by  expanding  a*. 

1  +x^ 

(1  +  xy 

Put  \  +  XzizZ. 


5.     To  integrate  dy  =  ^^    ^  e* .  dx. 


.•.a:=0-l,    l+a;2=l+s2-23+l=22_22+2,    dx=zdz,   e'=e'-\ 

or  by  integrating  the  last  term  by  parts 

y  =  Lre^-2/  —  dz-2—+2—dz]=e^-'^2 -^  0 


CHAPTER   VIII. 

TRIGONOMETRICAL    AND    CIRCULAR    FUNCTIONS. 

63.  S\nce  the  tangent,  cotangent,  secant,  cosecant,  versed-sine,  and 
coversed-sine,  can  all  be  expressed  rationally  in  terms  of  the  sine 
and  cosine,  it  will  only  be  necessary  to  investigate  formulae  for  the 
integration  of  expressions  involving  sines  and  cosines. 

64.  Prop,  To  obtain  a  formula  for  diminishing  the  exponent  m  of 
sin  a;,  in  the  general  integral 

y  =  /sin'"ar .  cos"2r .  dx^     when  m  is  an  integer. 

Put  cos»a: .  sin  x  .dx  =  dv,     and     sin'^-^a;  =:  u  ; 

cos**    a? 

then      V  = .-»     and     dii  =  (m  —  I)  sin"»~2^ .  cos  a? .  dx. 

n  +  I 

* 

and  by  the  formula  for  integration  by  parts 

/••  «          «'    J           sin"»-ia:.cos«+i2;  ,  m  — 1        ^  „         «+•>    j 
y =fsm^x .  cos"a;  .dx= 1 ——jsm^-^x.cos^^^x.dx. 

But  cos«+2a:  —  cos^a; ..  cos^a;  =  (1  —  sin^a;)  cos^a:? 

sin'"-^a: .  cos"+^a; 


y  =  - 


rA+  1 


A /sin'^2^. cos*a: .  dx — :  /sin*"ar .  cos*a; .  dx. 

Transposing  the  last  term  and  reducing,  we  obtain 

,  sin'"-^a: .  cos"+^a;  ,  m— 1  ^  .       „  , 

fsiTi^x .  cos"a;  .dx  = 1 jsvd^^H  .  cos"ar .  ax. 

m  +  n  m-\-n 


TEIGONOMETRICAL  FUNCTIONS.  299 

And  similarly, 

^.       „  ,  sin"»-3^.cos"+iar  .     m  — 3     ^.       .         ^     , 

fsmr^^x.Qos'^x.dx= ; — -fsm^'^x.cos^x.dx. 

m  -{-  n  —  2         m-{-n—2 

y..  «._A  -     J  sin"'-5;P.cos'»+ia:  ,     w*  —  5     . .  ^,-         ^     . 

/sin'^^a;.  cos^x.ax= 1 -fsm^'^-^x.cos^x  dx^ 

m -f- n  —  4         m-^n—4: 

&c.  &c,  &;c. 

Hence  by  successive  substitutionsf 

-,  cos«+ia;  r  .       ,      .        m  —  1 

y  =  /sin*"a; .  cos^a;  .ax  = ; —  sm'^-^ic  H ; sin"^^^ 

m  4- ^  m-\-n  —  2 

(m— l)(m— 3)         .       .         o 
(m+n— 2)(m+w— 4) 

(m-l)(m-3)(m-5) 4  or  3 

""       (w+n— 2)(m+7i-4)(m+7i-6)...(w  +  3)  or  (w-f-2) 

X  sin^ic     or     sin  a:] 

(m— l)(m— 3)(w— 5) 2  or  1 

(m  +  n)(m-fw— 2)(m  +  w— 4)...(/i-f3)  or  (^-{-2) 

X  /sin  ir .  cos"a: .  dx     or     fcos^x  .dx (JS'). 

65.  This  formula  renders  the  proposed  integral  dependent  upon 
that  of  the  form 

sin  X .  cos"a: .  dx     or     cos^a; .  dx, 

according  as  m  is  odd  or  even,  the  effect  of  the  formula  first  ob- 
tained being  to  diminish  by  2  the  exponent  m  of  sin  x,  at  each 
application. 

Also  the  first  of  these  two  final  forms  is  immediately  integrable 
by  the  rule  for  powers  :  for 

7/         X  cos^+^a;       ^ 

/cos"a: .smx.dx  =  —  /cos"a; .  a(cos x)  = —  -f  C. 

''  ^         '  n  ■{•  I 

Hence  we  have  only  to  obtain  a  formula  for  the  integration  of  the 
jrm  co^^x.dx,  in  order  to  effect  the  complete  integration  of  the 
proposed  differential. 


300  INTEGRAL    CALCULUS. 

66.  Prop.  To  integrate  the  form  dy  =  cos^a; .  dx  where  n  is  ac 
hiteger. 

Put  cos X .dx  =:  rfv,     and     cos"-*^a;  =  u  ; 

then  t;  =  sin  a:     and     du  z=:  —  (;t  —  1 )  cos^'-^ar  sin  x  dx. 

Hence  by  substitution  in  the  formula  fudv  =z  uv  —  fvdu,  we  obtain 

/cos^x.dz  —  sin  x  .  cos"-i^4-  (w^ l)/cos"~2^ .  sin% .  dx 

=  sin  X .  cos"-^a:-f-  (?^  — l)/c()s"~2^(l  —coii^x)dx 

=  sin  a; .  cos^-^^^-  [ti  —  l)fcos"-^x  .dx  —  ()i  —  \ )/cos"a; . dz. 

Transposing  the  last  term  and  reducing,  we  get 

sin  X .  cos"-i^   ,   n  —  \   ^  „       ,  ,     .    .,    , 

fcos^x.ax  = j fcos^-^x.dx,  and   similarh 

n  n  '' 

„       _        sin  a; .  cos"-^^:  ,   n  —  3  ^         ,     _ 

jcos^~''X .  ax  =z 1 -fcos'^-^x.dx. 

n  —  2  7i  —  2 

sin  X .  cos**-5a;      w  —  5  ^         „     , 

/cos^-^a; .  dx  = 1 -fcos^-^x.dx, 

n  —  4  n  —  4 

&c.  &c.  &c. 

Hence  by  successive  substitutions, 

sin  aTp  ,       w  —  1 

y  =  fcos^x .  dx=z cos'*-'^^;-] r  cos^-^a; 

n   ^  n  —  2 

(„_l)(^_3)(«-5)...4or3       .  ^ 

•  •  •  •  4-7 — ^7 7(7- — ^TT r. ^cos2a:  or  cos  a;] 

(«— 2)(w— 4)(w— b)  ...  3  or  2  ■* 

+  S SvT ~. o- o /«^s x.dxoT  fdx.  .  {F). 

/t(/A— 2)(;i— 4) 3  or  2  ^    '^ 

This  formula  renders  the  proposed  form  dependent  upon  one  o( 
two  known  forms,  viz. ; 

/cos  a; .  rfa;  =  sin  a;  4-  C',     when  n  is  odd, 
or,  fdx  =  X  -{■  Cf     when  n  is  even. 


TRIGONOMETRICAL  FUNCTIONS.  301 

67.  The  two  propositions  just  given  effect  the  complete  integra. 
tion  of  siu^a; .  cos«a: .  dx,  when  m  and  n  are  integers,  by  first 
diminishing  the  exponent  m  of  the  sine,  and  then  the  exponent  n 
of  the  cosine.  But  it  is  often  preferable  to  reduce  n  first,  and  for 
this  purpose  we  require  the  following  proposition. 

68.  Prop,  To  integrate  dy  =  sm^x .  cos"a; .  dx^  by  first  dimin 
ishing  the  exponent  n  of  the  cosine. 

If  in  the  formula  {JE),  we  make  a;  =  -  -tt  —  iCj,  w  =  Wj,  and 
n  !•=  Wj,  then 

sin  a?  =  cos  x^,     cos  x  =  sin  arj,     dx  =z  -^  dicj, 
and  by  substitution  we  shall  obtain 

/sin"*a; .  cos"2;  .dx  z=  —  /cos^iajj .  sin^^x^dx-^ 

sin'«i+ia:i  r      „    ,       ,    ^     i 

= ; cos^i-^rr,  +  &c.  , 

Wj  -(-  mj '-  -^ 

or  by  omitting  the  accents  and  changing  signs, 

,  .  ,         sin'^+^a;  _  ,      .        w  —  1  o 

fsm^x ,  cos"2; .  dx  = —  cos'*""-^a;  H ; cos*~3a; 

n  -^  m^  n-\-  m  —  2 

(H-\-m—2)(n-\-m—4) 

{n-l){n-S) (n-5) 4  or  3         

"^  {n-{-m—2)(n  +  m—4t){n-^m—Q),  .  (m  +  3)  or  (m-f2) 

X  cos^a;  or  cos  a;] 

"^(/i-|-m)(^+m-2)(M+w-4).  .(mH-3)  or  (m+2) 
X /cosar.  sin'":c.c?a;  or  / si n'".'c. c?ar.  ....   (G). 

But    fcosx.  sin'"^; .  c?a;  =  /sin'"a; .  c^(sin  x)—  — '■ +  C, 

^        ^      m  -\-  I 

which  will  be  the  required  form  when  n  is  odd. 

We  have  therefore  only  to  provide  a  formula  for  the  integration 

of  the  form  sin'"a: .  dx^  which  will  be  necessary  when  n  is  nvon. 


802  INTEGRAL  CALCULUS. 

This  may  be  readily  effected  by  substituting  in  formula  {F),  m 
1 
2 


for  n,  and  -  *  —  a;  for  x,  and  changing  the  signs.     Thus 


fsm^x.  dx  = [sm'^-^a;  -] sm'^-^a; 

•^  m     ^  m  —  2 

(wz — 2)(m— 4) 

(m-l)(m-3)(m-5)  ...  4  or  3   .  ,  .      ^ 

(m  —  2){m—4)(7n  —  b)  ...  3  or  2  -* 

(m  — l)(m— 3)(w  — 5)  .  .  .  2  or  1  ^.       ,          ^,  ,  „. 

+  ^^ /" — rr/^ — TT-^ r /sin  a;c?a;  or  /dx (H), 

69.  The  formulse  (G)  and  (/f)  effect  the  same  object  as  {B)  and 
(F),  reducing  the  integral  fs'm^x.cos^x.dx  to  one  of  the  known 
forms 

fdx  :=:  x-\-C^   /cosa:.c?a;  =  sina:-f  (7,  or,  /sina;.c?a;  =— cosa;4- C', 

the  exponent  m  or  n  which  is  first  reduced  being  an  even  integer, 
and  the  other  exponent  an  even  or  odd  integer. 

But  if  ?7z  be  odd,  {E)  alone  will  effect  the  integration,  whether  n 
be  an  integer  or  fraction ;  and  similarly,  if  n  be  odd,  ( G),  alone 
will  suffice. 

70.  Prop.  To  integrate  i;he  forms   dx,  and dx,    where 

cos"a;  sm'".r 

m  and  n  are  integers. 

By  the  formula  {JE)  the  first  of  these  forms  may  be  reduced  to 

dx  sin  X  .  dx 

or     , 

cos^a:  cos^a; 

dx  cos  x .  dx 

-. ,     or     : • 

sm'^^r  sm^'ic 


and  by  {G),  the  second  may  be  reduced  to 


-,  ^    fsinx .dx      cos-^+^ic     ^      .   Pcosx.dx          sin"-""**^*  ,     -, 
But/  -— = ^Cand  /  —:-- — = — r  ■♦   ^• 

Hence  there  will  remain  to  be  integrated  the  forms 

cos-'^x.dx (1),         and         sin-"a;  .dx (2). 


TRIGONOMETRICAL  FUNCTIONS. 


803 


Put  in  (1),     cosx.dx  —  dv,         and         cos-^-^a;  =  u, 
then         V  =  sin  a;,         and         du  =  {n  -\-  l)cos-"~22;  ^  gj^  x .  dx, 
and  by  substitution  in  the  formula  for  integration  by  parts, 

fcos—^x  .dx  =  smx.  cos""^^^;  —  (n  +  1)  /sin^ar .  cos-^-^^g  ^  ^gg 
=  sin  a; .  cos-^-^a:  —  (^  -h  1)  fGos-~^~^x .  dx  -\-  (n  •}-  \)  fcos'~^x .  dx. 

Transposing  and  reducing,  we  get 

/dx  sin  x  11       P  dx 

cos^+^^^r  ~  in  4-1  )cos'»+^a;       n  +  \J  (. 


X       {n  -\-  Ijcos""^^ 

sin  a;  .   to  —  2   /* 

n  —  1«/  ( 


cos"a; 


;  and  by  analogy 


/ClX                                           Olll   A>                                     lb     —    i<J       J              UJU  ,  ,^ 
=  7 r-. z — I / — ,  and  similarly 
cos"a;      {n  —  l)cos"~ia;       w  —  1«/  cos""^^;'  •' 

/dx  sin  a:  n  —  4i   P     dx 

cos'*"^^;  ~~  (n  —  3^cos"~^a:      n  —  3«/  cos"~*a; 


{n  —  l)cos"~i 

dx  sin 

cos'»~2^  ~  (w  —  3) 

&c.  &c, 


&;c.         Hence  by  substitution 


/dx         sin  a:  r      1 
cos^a:  ~  w  —  1  Lcos"~ia:      in  — 


(n— 3)cos"~% 
(n-2)(TO-4) 


{n — 3)(w — 5)cos"~^a; 
(w-2)(to-4)(w-6) 3  or  2 


4-&C. 


J 


+ 


^  r  dx 

1  / or  fdx (/). 

1 «/  cos  X  ^    ' 


(n— 3)(/i  — 5)(m— 7) 2  or  1  .cos^a;  or  cos  a:. 

(w— 2)(n— 4)(n-6) 1  or  0   P  dx 

\n-\){n-~^){n-'b)  .  .  .  .  2  or 

The  second  of  these  integrals,  fdx=zx-\-  C,  will  never  be  re- 
quired, because  its  coefficient  is  zero,  and  therefore  we  stop  at  the 
preceding  term.    For  the  first  we  have 

/dx    _  P  cos  X  .dx       P  cos  a; .  rfa;  _  1    /"cos  x,dx      1    /"cos  x .  dx 
cos  X     J      cos^a;     ~J  1  —  sin^a?  ~~2«'   1  -f- sin  a;     2«/   1  -— sinas 

=  ilog(l+sin^)-ilog(l-sinx)+e  =  log[i±^-||]*+(7 
2sin(jr+ix)cos(l<-l:r)"'* 


=  log 


_2sin(i*-l:.)cos(i*  +  i.)_ 


804  INTEGRAL  CALCULUS. 

dx 


/dx 
.  ^  ,  replace  in  (/),  n  by  m, 


X  by  -"jr  —  ar,  and  y  by  z.     Then 


/dx    __        cosa;  r      1 
sin'"a;  ~       w  —  1  Lsin'""^ 


w— 2 


ic       (m— 3)siu'"-3;5 


(m— o)(w  — 5)  siii'"-^a; 


(??i— 2)(m— 4)(77i— 6)  ...  3  or  2  "1 

(m— 3)(m— 5Xwi— 7)  ...  2  or  1 .  sin'^^o;  or  sin  a:J 

(^_2)(m-4)(m-6)  ...  1  or  0  /•  rf^     ^ 

The  second  integral  has  a  coefficient  equal  to  zero,  and  therefore 
will  never  be   used.      For  ti:e  first  we  have,  by  replacing  x  by 

-  <  —  a;  in  (/j) ,  and  changing  signs 

i 

/- —  =  —  loff  cot  Tzx  =z  log ; —  =  loff  tan  -  x  -f-  C. 
sin*              ^2             ^1  '=2 

cot -a; 

dx 
72.  Prop.  To  integrate  dy  = -^  where  m  and  n  are 

integers. 

Since  sin^a:  +  cos^a;  =  1. 

/dx  P{s\n^x  4-  co»^x)dx 

sin'"a; .  cos"a;  ~  *^        sin'"^: .  cos'»a; 

/dx  r  dx 

sin"»-2a; .  cos"a;      •/  sin'"a; .  cos^-^a; 

/(sin^a;  +  cos2a;)c?a;        /^(sin^a;  -f  cos2a;)(fa; 
gjj^m-2^ .  cos^a;  J      sin'^a; .  cos"-^^ 

/dx  r  2dx  I     /*  ^^ 

sixi^-^x .  cos"a:      *^  sin"*~^a; .  cos^-^a;      *^  sin'^a; .  cos*~*af  * 

and  by  continuing  to  introduce  the  factor 

sin^a;  +  cos^a:  =  1, 


TRIGONOMETRICAL  FUNCTIONS.  805 

we  obtain  finally  one  or  more  of  the  following  known  forms 

x.dx      rQ,Q)'$,x.dx       Psmx.dx      Pcosz.dx 


/dx       p  dx        Psmx.dx      Pcosx.dx       Psmx.dx      P( 
sin'^jr  «/  cos^aj  •/     cos"a;      «/     sin'^ic       •/      cos  a:       «/ 


smic 


Applications  of  Formulce  (E),  (F),  (G),  (H),  (I,)  and  (K). 

73.   1 .  To  integrate     dy  =  sin^^; .  cos^a; .  dx. 

Here  m  =  5,  and  n  =  5,  and  since  both  are  odd  we  may  apply 
(E)  or  ( G)  with  equal  advantage.     Employing  (U)  we  have 

y  = r^  [sin%'  +  -  sin^a;]  +  -^-—  fsmx,  cos^a; .  dx 

f»Qg6/g  11 

[sin*a;  +  -  sin^a:]  —  cos^o;  +  (7 


10    *-  '2         -"       10.6 

cos^a^r  ,1.9      ,    n  ,    ^ 

2.  c/y  =  sin% .  cos^a: .  dx. 

Here  m  =  6,  yi  =  3,  and  since  t^  is  odd  we  apply  (G), 

su-i'x  r      2  1    I   ^  /•  •  fi      ^         s"^''^ /      9      ,  2\  ,   ^ 

.' .  y  —  [cos^icj  +  -  /cos  a; .  sm^a; .  dx  =  — — - 1  cos^a:  +  - 1  -f-  C/. 

3.  (;?y  =  sin^ic .  dx. 

In  (^)  make  m  z=z  6. 

cos  a;  ,5.,     ,5.3.      ^5.3.1       ,„ 

.  • .  y  = ^  [sinSa;  +  -  sm^a;  +  ^-^  sm  a;]  +  ^   ^   ^  x  +  C. 

4.  c?y  =  sin^a; .  cos^ar .  dx. 
In  (B)  make                 m  =  8     and     n  =  6. 

cos^a:..  -     ,    7    .  ,     ,      7.5      .  ,     ,     7.5.3      .      ^ 

y  =  -  IT  t^^^  ^ + 12  '^^  "^  + 12710  '^"  ^ + T2no:8 '"'  ^^ 

7.5.3.1      ,      ,      , 

+  14. 12. 10. 8-^^"^  ^•^^' 
20 


806  INTEGRAL  CALCULUS. 

and  by  applying  (F)  to  the  last  term,  we  get 

cos':r  7  7  7 

sin  a:  r      ,      ,   5       ,     ,    15  ,    ,      5a;     .     _ 

+  ^yg    [0OS»^  +  -  C0S3X  +  _  COS  *]  +  ^^  +   a 

5.  dv  =  - — T-  f^^- 

In  (-£^)  make  w  =  5     and     n  =  —  2.     Then 

—  1     r  .  .      .   4    .  ^  ,       4  .  2  /*  sin  a: .  c?a; 

y  =  ^ [sin*a;  +  -  sin^:?;]  -\-  — —  / — 

3  cos  a;'-  1  -^       3.1«/      cos^a; 

""        [sin*a;  +  4  sin2a;  —  8]  -f  C. 


3  cos  a; 

t/a: 


6.  rfy 


In  (JST)  make  m  =  5.    Then 


_  _  ^Qs^  r    1       ,        3     1      3J^    r  dx 

~~  4     Lsin*a:      2  sin^a^J      4 . 2  •>'  sin  a; 

cosa^ri       .        31.3,  ^.^ 


7.  c?y  = 


cos^a; 
In  (/)  make  /i  =  6.     Then 


^  sin  .r  r    1  4    J_   ,   4    2    1] 

^  5     Lcos^a?  "^  3  *  cos^a;       3  *  1  cos  a;J  "*"     * 


8.  dy  = 


sin*a; .  cos^a; 
Introducing  the  factor  sin^a;  +  cos^ar,  we  obtain 


/(sin^a;  4-  co^'^x)dx       P        dx  P  dx 

sin*a: .  cos^a;       ~J  sin^a: .  cos^a;      «/  sin^a; 

/dx  n  dx  r  dx 

cos^a;     »/  sin^a?      «/  sin*a; 

\    Lsin^a:       sin  a:  J 


=  tana:— cot  a; • 


TRIGONOMETRICAL  FUNCTIONS.  807 

74.  When  m  =  —  n^  formulae  (B)  and  {G)  cease  to  be  applica- 
ble,  but  we  then  have 

V  =  I dx  =  ftan^x  .dx     or     y  =  I  —. dx  =  I  cofar  dj: 

^      J  cos«a;  "J   sm«a;  J 

To  integrate  the  first  of  these  expressions,  put  sec'^  —  1  for  tan^ 
and  in  the  second  put  cosec^  —  1  for  cot'^.     Thus 

/tan^a; .  dx=JX>w?x .  tan"-^^; .  dx-=.f%^(^x .  tan^-^ar .  dx—ftan^-'^x.di 
1 


tan"-^a;  —  ^^tan^-^a;.  c?a; 


tan"-^^;  —  /  (sec^a;  •  -  1 )  tan*~*a; .  dx 

tan^-^a; tan^-^a:  H-Ztan^-^a; .  dx 

n  —  I  n  —  6 

tan"-^a: tan'*-^^  -| _  tan^-^a;  —  &c, 


n  —  1  w  —  3 

the  last  term  being 

'  sin  a: .  dx 
cos  a; 

when  n  is  odd  or  /c?a;  =  a;  +  C  when  n  is  even. 
cos"a:c?a; 


/sin X   dx 
—  =  —  log  cos  a;  +  (7  =  log  sec  a;  +  0 
cos  X  ^ 


CI.     .,     1                          r  QOS'^XdX  C  7 

Similarly,  /  — ■. =  /  cot"a; .  dx 


cof'-ia;   ,    cot«-3a:       cot«-^a:   ,    . 

= H +  &c. 

n  —  1         n  —  6         n  —  5 

The  last  term  being  /cot  x  .dx  =.  log  sin  a;  +  C^  or  fdx  ■=:z  x  -\-  C. 

75.  When  the  proposed  form  is  /sin'"a:.cos'*arc?a;,  in  which  m  and  n 
are  integers,  the  integration  may  be  conveniently  effected  by  con- 
verting the  product  sin'»a; .  cos"a;  into  a  series  of  terms  involving 
sines  or  cosines  of  multiples  of  x.  The  integration  can  then  be 
perfoi-med  without  introducing  powers  of  the  sines  or  cosines. 

The  proposed  transformation  can  always  be  accomplished  by  the 


308  INTEGRAL  CALCULUS. 

repeated  application  of  one  or  more  of  the  three  trigonometrusal 
forraulfie. 

sin  a  cos  6  =  -  sin  (a  -f-  6)  -j-  -  sin  (a  —  6), 

sin  a  sin  6  =  -  cos  (a  —  b)  —  -  cos  (a  +  h\ 

COS  a  COS  6  =  -  cos  (ct  —  6)  +  ^  cos  (a  +  6). 

To  illustrate  this  process  take  the  following  example. 
dy  =  sin^o; .  cos^xdx 


.        /  .  NO         .       /sin  2ar\2 

sm-^ic .  cos^^c  =  sm  x  (sm  a; .  cos  a;)^  =  sm  a:  I  — - —  I 

1    .       /I  —  cos  4a;) 

=  -  sm  a: 
4 


(1  —  cos  4a;\       1    .  1    . 
1  =  -  sm  a;  —  -  sm  a; .  cos  4a: 
2/8                o 


=  -  sin  a;  +  TTT  sin  3a:  —  -—  sin  5x, 
o  lb  lo 

.  • .  y  z=  /  I  -  sin  a:  +  T^  sin  3a:  —  —  sin  5a:  Jia; 

=  —  -  cos  a:—-—  cos  3a:  +  —  cos  5x  -\-  C. 
o  48  oO 

76.  Prop.  To  integrate  the  form  dy  =  b°^ .  sin"a: .  dx. 

Put     sin X .dx  =  dv,  and  ft"^sin"~^a:  =:  u,  then  v  =  —  cos x, 

and     du  =  (n  —  l)b°^^in^~'^x .  cos xdx  -\-  a  .  log  b  •  6«^sin"~^a: .  dx. 

.'.  y==:f  b'^^sm.^x.dx=:— b°^^sm^~'^x. cos x-{-{n— I) fb^^s'm^-^xcos^xdx 

+a.  log  6./6"sin"~^a:.  cos  x.  dx. 

But,  by  applying  the  formula  fudv  =  uv  —  fvdu  to  the  last  integral, 

making  sin*-'a:.cos x.dx  =  dv  and  6*^*  =  u,  we  get 

/6'»^sin"'-ia;.cosa;.c?a;=-sin"a:.6** a  log  6 /sin"a? .  6" .  cto, 

n  n 

•nd,  by  replacing  cos^a;  by  1  —  sin^a:,  we  have 

/'5axsinn-2^  cos^a: .  dx  =  rb''^sm*-^xdx  —  /6«=^£in"a; .  dg.     ■■ 


TRIGONOMETRICAL  FUNC'I10nC>^  ^'>    309  /     , 

ff-^nce,  by  substitution,                                            ^^^^^^"^^'^  /'  ^ 

Jh^^ .  sin"« .  aa:  =  —  o°^sm"-ia; .  cos  a;  H ^-  o«*Wn"t^   i  ^         O  > 

(alos^y  ^^"'^ 

-  ^^ ^-^  /sin*^ .  6«^6?a;  +  (w  —  1 )  / 6«^sin«-2a; .dx       ^V_  * 

—  (n  —  V)  fh^^^\X)J^x  .dx. 
\  'ansposing,  colie^-ting  like  terms  and  reducing,  we  obtain 

/  i-^sui-a: .  dx  =  /^|^g^N2_{.,^2  (<^  ^^g  6 .  sm  a;  -  n  cos  x) 

-i-  7-'- --r.^?     o  /^°^siii"-2^ .  c^a;  ...  (X). 
B^  repeated  applications  o^  (Z)  we  obtain  the  final  integral. 

lyax 

j'h^^dz  =  — ; — r  +  C,  whv>ii  n  is  even  ;  and  when  n  is  odd, 
a  log  0 

J  b'Hhi X .  dx,  which  is  given  bv  (Z)  without  an  integration,  since 
the  lasS  ler.n  then  contains  the  factor  n  —  1=1  —  1=0,  and 
therefore  tha^  term  disappears. 

77.  Prop.  lo  integrate  the  form  d^  -  •   ^'^^^cs^o; .  dx. 

Put  a?  =  a;^  —  -  r,  then  cos  x  =  sin  a: .,  sir:  ;:• .-  —  cos  ajj, 

i»*  _-  5««i.o  ^    ^    dx  ~  OvVj. 

,-i^"  -,       .         ,        ^        6«^isin»-^ar,.  ,    . 

^•,yz=zb  ^    /6«^ism«a;  d>;  n=- -7^-i ttt-, — ^('^J  Vr^  fma?,— ncosa;,) 

^  "^  '     *        (a  log  0)2+^2    V        .  1  ./ 


■ra 


n(n--l)b  ^      ^,       .       o     7  11  •.    .• 

4-  -\ -^- f  I'^^is^L'^^x-.dXi,  and  by  sausiic'uiGn, 

^(alogZ>)2-f7l2-^  1       1'  J  » 

fb'^'cos*x  .  dx  =  -7—; r,^  .  --  (a  log  6 .  cos  a:  -^  r  &!ia  j<J 

(alog6)2-f.n^  ^       "^  -^ 

n(7i  —  1)       ^,  „       ,  ,-,,^ 


SIO  INTEGRAL  CALCULUS. 

lax 

Here  the  final  intem-al  will  be  Jh'^^dx  =  — j +  (7,  whei.   n  is 

^  a  log  6 

even  ;  and  when  n  is  odd,  f  b'^^c.os  x .  dx^  to  which  (if)  applies  with, 
out  an  integration. 

1.  To  integrate  dy  =  ««^ .  cos  x  .dx. 

In  {M)  make       6  =  e,  n  =  1,  log  6  =  log  e  =  1.     Then 

y  =    ^    .       [a  cos  a;  +  sin  a;J  +  C7. 

2.  dy  :=  e^ .  sin^a: .  (fa;. 

In  (Z)  make    6  =  e,  a  =  1,  »  =  3,  log  6  =  I.     Then 

e^ .  sin2a;  ^  .  «  .,.3.2,. 

y  =  yTTs^  L^'n  a;  —  3  cos  a;]  +  ^       ^^  fe^'smx.dx 

1  fi     1 

—  e^  [sin^a;  —  3  sin^a; .  cos  a;]  +  tt:  • «  e*(sin  a:— cos  a;)-f'  C. 


10      ^  -^    '    10   2 

1_ 

10 


or,   y  =  —-  e*[sin3a;  +  Scos^a;  -f-  3sin  x  —  6cos  x'\  +  C. 


8.  c?y  =  e-<"sin  kx.dx  =  - e-«*sin A:a? .  c?(A;a:). 

In  (Z)  make     6  =  e,     a?  =  ^a;,     a  ==  —  -?     Then 

er'^'^ia  sin  A-a;  +  ^  cos  A:a:) 
^  k'^  +  a'^ 

78.  Prop.  To  integrate  the  form  dy  =  X.  sin-^a:.  c?ar,  in  which  X 
is  an  algebraic  function  of  x. 

Put  Xdx  =  dv,         and         sin~^a:  =  u ; 

Uien         V  =  fXdx  =  Xj,         and         du  = 


.  /»    X-^dx 

and  the  proposed  integral  is  thus  caused  to  depend  upon  anrjther 
whose  form  is  algebraic. 


TRIGONOMETRICAL  FUNCTIONS.  BIT 

79.  Prop.  To  integrate  the  form  dy  =  Xcos-^x.dx^  in  which  X 
IS  an  algebraic  function  of  x. 
Put  Xdz  =  dv,        and        cos-'o;  =  u ; 

dx 

then        v  =  /Xcilr  =  X^       and         c?«^  = ; 

.  • .     y  =  Xcos-^a;  -f    / — -^^ ,  an  aleebrais  form. 

./    -/l  -  a;2 

Cor.  The  same  process  will  apply  to  each  of  the  forms 
Xtaxr'^xdx,     X  cot^^zdx,     Xsecr'^xdx^  6zc., 
since  the   differential  coefficients  of    tan-^ar,    cot^^x*,     sec-^a;,  &c, 
are  all  algebraic. 

1.  dt/  =  ar^sin'^o: .  dx. 

Here     X  =  x^,         .-.    X^  =  fXdx  =  fx^dx  =z  ^ 


3.  t/y  =  -—- — 1  •  tan  -*«. 


x'^dx 


Put     c?v  = =dx  —  — - — ;:,        and        u  =  tan~^«, 

1  4-  a;2  1  4-  a:^ 

.  * .     V  =  ar  —  tan-^a;,         and         du  -z 


l+a;2 
xdx  f  tan~ia; .  dx 


,  xo         /*   ^'"^     .     /"tan" 
.  • .  y  =  a:  tan-^a:  -  (tan-i;r)2  -  J  j^p^2+  7  — 


■^-x^ 


=  ar  tan-la;  -  (tan-W- ^  log(l  +  x^)'\-\{XmrHf  ■{■  C. 
1 


tan"-iar(a;  —  -tan"»a;)-  log  yl-l-a^  -r  0, 


CHAPTER    IX. 


APPROXIMATE    INTEGRATION. 


80.  When  a  given  differential  cannot  be  reduced  to  a  form 
exactly  integrable,  we  may  expand  the  differential  coefficient,  either 
by  Maclaurin's  theorem,  by  the  common  binomial  theorem,  or 
otherwise ;  then  multiply  by  dx^  and  finally  integrate  the  terms  suc- 
cessively. If  the  resulting  series  be  convergent,  a  limited  number 
of  terms  will  give  an  approximate  value  of  the  integral. 

81.  This  method  may  also  be  employed  with  advantage,  when  an 
exact  integration  would  lead  to  a  function  of  complicated  form. 
And  the  two  methods  can  be  used  jointly  to  discover  the  form  of 
the  developed  integral. 

EXAMPLES. 

82.  1.  To  integrate    dy  =  — — —  dx,  in  a  series. 
Expanding  by  actual  division,  we  have 

1  —  a:  -}-  a;2  —  ic^  +  ic*  —  &c 


\^x 
.  • .   y  =  /(I  _  a;  4-  a?2  _  a;3  -f  a:*  —  ^Q.)dx. 

the  required  series. 


APPROXI^IATE    INTEGRATION.  313 

Again  J  ■— _—  dx  —  log(  \  -\-  x) -{■  C^ 

. '  .     log(l  +  x)—  X ^^  +  r  ^^  —  -  x'*'  -{-  -  x'^  —  &C.-J-  c, 

^^  2  a  4  5 

whore     c  =  C  —  C-^. 

But  when     a;  ==  0,     ]og(l  -\-  x)  =  \og\  —  0,     .-.0  =  0, 

.  • .     log(l  -\-  x)  =  x  —  -  x"^  -\-  -  x^  —  -  x^  -{■  -x^  —  (Szc. 
^  o  4  o 

a  well  known  formula. 

2.  c/y  ==  ;r*(l  -  x-^ydx. 
Expanding       (1  —  x"^)       by  the  binomial  theorem, 

(l-.^)*=l-i.^-i.^-l.e_-|-.s_&„. 

.-.  y  =  fx^{l -g ^^  -  8 **  -  1^  =^°  -  ni ^°  -  &c.)<&. 

=  3^-7^-44^-125^  -mi"  -&c.+  a 

3.  dy=z—:=,' 

= ^  -  o  ^' + 2^"' -  ari^o"' +  *'''•  +  ^- 

dx 


But  /*-^^=  =  log(a:  +  vTT^)  +  Ci. 

•/-j/l  -\-  x^ 

.-.  log(*+v/I+^)=x-^x3+-!^;r»-H^:r'+  &c.+  (7-(r.. 


814  INTEGRAL    CALCULUS. 

Now  when   xzzzO,  \og(x+^+x^)=\ogl=0.     ..C—C^—d. 
_^— ^—  1  1  ^  1  ^  'i 

.  •  .  log  (^  +  ^l+X^)=.X-  ^X^  +  ^ri-=  X^  -  5^-^  X''  +  &C. 


2.3       '   2.4.5  2.4.6.7 

1+"^ 


dx 
4.  To  integrate  dy  =  —— — ^  both  in  ascending  and   descending 


powers  of  x. 

— — -=l-a;24-a:*-a;6+&c.  and  — —-  =:_-—_+  — ~--+&c. 
l-\-x^  x^-\-l       x^       x^       x^      x^ 

/dx 
— - — -  =  tan-la;  +  C  =  f  il  ~  x^  ■\-  x^  -  x^  k.Q.)dx 
\  -\-  x^  ^  ' 


=  X  — -x^  ■\-  -x^  —  -  x'^  -\-  &c.  +  C. 

dx 


The  two  results  become  equivalent,  by  selecting  the  constants  C 
and  Ci  such  that  Cj  —  (7  =  -  -jr. 

For,  the  first  series  =:  tan-^ic  +  0. 
And  the  second     "      =  —  tan-^  -  +  C,  =  —  cot"' a:  +  6\. 

X 

.  • .  In  order  that  the  two  series  may  be  equal,  we  must  have 
tan-^a;  +  (7  =  —  cot-^a;  +  Cj, 

or  tan-^a;  +  cot-^a;  =  C'l  —  (7,     or    - 1  =  Cj  —  C. 


5t  dy  —  ^  _  —  dx. 

Expanding  the  numerator  we  have 

(1  -  e2a;2)*=  1  - 1  e^x^ -\-  J—  e^x*  -  J  \^ ^  eH^  &;c 
"^  '  2  2.4  2.4.0 


APPROXIMATE   INTEGRATION.  816 


11  13  dx 


2.4  2.4.6  'yir=^ 

/- and    have    been 

/I  —  x^ 

already  integrated  in  the  chapter  relating  to  binomial  differentials. 

We  might  also  expand  (1  —  x"^)  by  the  binomial  theorem,  then 
perform  the  multiplication  indicated,  and  finally  integrate  the  terms 
in  succession.     Adopting  the  first  course  we  have 

y  =  sin-^a;  +  h  ^^  (o  ^V^  ~  ^^  ~  9  shi-^a:! 

83 •  Prop.  To  obtain  a  series  which  shall  express  the  integral  of 
every  function  of  the  form  Xdx,  in  terms  of  X,  its  differential  co- 
efficients, and  X. 

Put      X  =z  u,  dx  =  dv  :     then     du  =  -^-  •  dx,     and     v  =  x, 

dx 

Now  substituting  in  the  formula  fudv  =  uv  ~  fvdu  we  get 
fXdx  =  Xx  —   I  -^  •  xdx. 

Next,  put  -J-  =  i^     and     xdx  =  dv^ 

dx 

then  du  =z    ,^'dx     and     v  =  — -—  x^, 

dx^  1.2 

rdX    ,        dX      x'         pd'^X     x^     , 

•••y^^^"=-^TT2-y^TT2^"- 

^.    .,    ,       rd^X     x^    ,        d^X      x^  pd^X      x^      ,    ,     , 


By  substiution 

^^,       ^       dX     x^    _^  d^X      x^  d?X        a^        ,  0       ,   n 

fXdx=Xx—  -—  . -— -  4-  -—  ♦     ^.   ^  —  -i-Y  • ,  ,^  .3  ^  -J-  &c.  4-  0. 
dx     l.'Z       dx^    1.2.3       dx^     1.2.3.4 


816  in:egrai  calculus. 

This  formula,  called  Bernouilli's  series,  shows  the  possibility  of 
expressing  the  integral  of  every  function  of  a  single  vaiiahle,  in 
terms  of  that   variable,   since   the   several    differential    coefficients 

-T—>  —i-^->  &;c.,  can  always  be  formed.     But  the  series  is  often  diver- 

gent,  and   then  of  no  use  in   giving  the  value  of  the  integral  ap- 
proximately. 


CHAPTER   X. 

INTEGRATION    BETWEEN    LIMlTb    AND    SUCCESSIVE    INTEGRATION. 

84.  The  integrals  determined  by  the  methods  hitherto  explained 
are  called  indefinite  integrals,  because  the  value  of  the  variable  ar, 
and  that  of  the  constant  (7,  both  of  which  appear  in  the  integral,  re- 
main undetermined.  But  in  applying  the  Calculus,  the  nature  of  the 
question  will  always  require  that  the  integral  should  be  ta\o»i  be- 
tween given  limits.  Thus,  suppose  the  integral  to  originate,  [yr  its 
value  to  reduce  to  zero)  when  x  ^z:  a:  this  condition  will  a/:  the 
value  of  the  constant  C.  Then,  to  determine  the  value  of  tbo  e^-'tire 
or  definite  integral,  we  replace  x  by  6,  the  other  extreme  vali  &  of 
the  variable. 

Ex.  To  integrate  dy  =  Sx^dx,  between  the  limits  x  =  x^  and  x  .   te^ 
y  =  fSx^dx  =  x^  -\-  C.     But  when     x  =  x^,     y  =  0. 
.  • .  0  =  arjS  4-  (7    and     C  =  —  x^\ 
and  by  substitution  in  the  indefinite  integral 
y  =z  x^  —  x-^^. 
Now  make  x  =  org,  and  there  will  result 
y  =  x^  —  XjS, 
the  complete  or  definite  integral. 


INTEGRATION  BETWEEN  LIMITS.  317 

A  slight  examination  will  show  that  the  desired  result  will  always 
be  obtained  by  substituting  in  the  indefinite  integral  for  the  variable 
a?,  first  the  inferior  limit  x^^  and  then  the  superior  limit  X2,  and  then 
subtracting  the  first  result  from  the  second.  In  these  substitutions 
the  constant  C  may  be  neglected,  since  it  will  disappear  in  the 
subtraction. 

85.  The  integration  ofSx'^dx  between  the  linAits  arj  and  X2,  when  x^ 
■s  the  inferior  limit,  or  that  at  which  the  integral  originates,  and  X2 
the  superior  limit,  is  indicated  by  the  notation. 


/ 


Sx^dx. 


86.  The  precise  signification  of  this  definite  integral  will,  perhaps, 
be  better  understood  by  the  aid  of  the  following 

Prop.  The  definite  integral   L  Xdx,  (where  X  is  a  function  of  x^ 

which  does  not  become  infinite  for  any  value  of  x  between  the  limits 
X  =  a  and  x  =  b,)  is  the  limit  of  the  sum  of  the  values  assumed 
by  the  product  Xh,  as  x  is  caused  to  increase  by  successive  equal 
increments  (each  =  h)  from  x  z=  a  to  x  =z  b;  the  value  of  h  being 
continually  diminished,  and  consequently  the  number  of  these  incre- 
ments being  indefinitely  increased. 

Thus,  if  Xq  X^  X2  X^  . . .  Xn^i  be  the  values  assumed  by  X,  when 
X  takes  successively  the  values  a,  a-\-h,  a-j-2A,  a-{-Sh, . . .  a+(w  — ])A, 

then  will  /a  Xdx  be  the  limit  to  the  value  {Xq-^X-^-^-X^..  .-\-Xn~i)h, 

provided  nh  z=z  b  —  a,  and  h  be  diminished  indefinitely. 

Proof.  Let  x  and  x  -\-  h  he  any  two  successive  values  of  x,  and 
denote  by  Fx  the  general  or  indefinite  integral  f  Xdx. 

Then  by  Taylor's  Theorem, 

rr/     .    X.N        r.     ,  ^^^  ^    .   d-'P'^  ^'     ,   d^Fx     h^       .    , 


818  INTEGRAL  CALCULUS. 

which  may  be  written,  F{x  -\-  h)  =  Fx  +  Xh  -{-  Ph"^,  ...  (1),  where 
P  is  a  function  of  x  and  h. 

Suppose  the  difference  6  —  a  to  be  divided  into  n  equal  parts, 
each  equal  to  A,  so  that  b  —  a  =z  nh. 

Now,  putting  successively  a,  a  +  ^5  a  +  2A  .  .  .  a  +  (w  —  1  )A  for  ? 
in  (1),  and  denoting  the  corresponding  values  of  P  by  Pq,  -^i,  &c., 
we  get 

F{a  J\-K)^Fa-\-  X^h  +  P^h"^ 

F(a  4-  2A)  =  F[{a  +  h)  -\-  h]  =  F{a  +  A)  -j-  X^h  +  P^h^ 

F{a  +  3A)  =  F[{a  +  2h)  -i-  k]  =  F  {a -[- 2h)  -{-  X^h  +  Pg^^ 

&c.  &;c.  &c. 

F{a-\-nh)  =  F[{a-\-{7i-l)h)  +  h]=F[a  +  {n-l)k]  +  Xn-,h-\-Pn-,h? 

adding  these  equations,  and  omitting  the  terms  common  to  both 
members  of  the  sum,  there  results 

F(a  +  nh)  =  Fa-{-  h(X,  +  X^  +  Xg  .  .  .  .  +  Xn-,) 

+  h^P,  +  P^  +  Pg  .  .  .  .  +  P„_3). 

But,  since  every  value  of  X  is  finite,  none  of  the  values  of  P  will 
become  infinite.  If,  therefore,  we  denote  the  greatest  value  of 
P  by  P,  we  shall  have 

Po+Pi-{-P2'-'-^-Pn-i<Pn,andsmceF{a-\-nh)=:Fb,iindnk=b—a. 

.-,  Fb-Fa-h{X,-\-X^  +  X^.,.-\-  Xn-,)  <{b-  a)P .  h. 

But  6  —  a  and  P  are  both  finite,  and  therefore  by  diminishing  A,  the 
second  member  can  be  rendered  less  than  any  assignable  quantity. 
Hence  Fb  —  Fa  must  approach  indefinitely  near  to  equality  with 
h{XQ  -f-  -STi  +  Xj .  .  .  .  -|-  Xn_i)  when  h  is  continually  diminished. 


SUCCESSIVE  INTEGRATION.  319 

Successive  Integration, 

87.  If  the  second  differential  coefficient  —rr  =  Xhe  given  instead 

of  the  first,  two  successive  integrations  will  be  required  to  deter- 
mine the  original  function  y  in  terms  of  x.  Thus,  multiplying  by  dx 
and  integrating,  we  get 

/-S-<^^  =  /^'^^'     or    %  =  fXd.^X,+  C,. 
Multiplying  again  by  dx^  and  integrating,  we  get 


f^  dx  =  fX^dx  -h  /  C^dx, 


dx 
or  y  =  Xj  +  C^x  +  C/g. 

d^t/ 
88.  Similarly,  if  there  were  given  -j-j  =  X,  three  successive  in- 
tegrations would  give 


y  =  x,  +  -^c,x^+c,x+a 


1.2 

d^y 
And  if  there  were  given        -7-^  =  X,       then 

(tx 


1.2.3...(w-l) 


'    1.2.3...(n-2) 
the  number  of  arbitrary  constants  introduced  being  n. 

89.  The  result  obtained  by  performing  the  above  integrations  may 

be  indicated  thus 

f^Xdx""  zzz  y : 

it  is  called  the  w**  integral  of  Xdx". 

90.  Prop.  To  develop  the  n'*  integral  f^Xdx^  in  a  series. 
Employing  Maclaurin's  Theorem,  we  have 


320  INTEGKAL   CALCULUS. 

"^  Laf^2  J  1.2.3..  (m4-2)'^  Lc/x-^  J  iT2:'3T:(:^~3y  ^     •  •  •  l^-I- 

The  terms  within  the  [  ]  are  the  arbitrary  constants  C\  C^  C3 . . .  (7,,, 
as  far  as  \^j' Xdx^  inclusive,  but  taken  in  an  inverted  onler. 

91.  Prop.  To  deduce  the  development  of  /""Xclx^  from  that  of  X 

By  Maclaurin's  Theorem,  we  have 

and  this  may  be  converted  into  the  series  [72]  by  multiplying  each 
term  by  ar",  then  dividing  the  successive  terms  l)y  I  .  2  .  3  .  .  .  n. 
by  2 .  3 .  4  .  .  .  (^i  +  1),  by  3  .  4  .  5  .  .  .  (/i  +  2),  &c.,  and  finally 
annexing  terms  of  the  form 

C^x-^  C^x--^ 

1.2.3  ...  (/i-  1)'     1.2.3..  .  (m- 2)'  "• 


/*     dx* 
- — ^ — — 


Here  X^iX—x')  *=1  +  1^2  +  1.  -  ^^  +  L  t  •  - -^^  +  &o. 

2  2  4  2   4    h 

Also  «  =  4.  Therefore  multiplying  by  .1*  and  dividing  successively 
by  1.2.3.4,  by  3.4.5.  O,  &c.,  and  finally  annexing  the  terms 
containing  the  constants,  we  get 


J  ^/\Z^    *"^    1    "^1.2'^1.2.3"^1.2.3.4"^2.3.4.5.6 

4-  &c. 


1.3a;8  .  1.3.5^i'> 


2.4.5.0.7.8   ■    2.4.0.7.8,9. 10 


SUCCESSIVE  INTEGKATION.  821 

2.  What  curves  are  characterized  by  the  equations  -r-^  =  0,  and 

(Py 

—y  =  0,  respectively  1 

\..U         g=0,     then     /S..=|=C.. 

.  • .  I  -j-dx  =  f  Cydx^     or    y  =  C-^x  -f  Cg,     a  straight  line. 

2a.if      g  =  o.  then  /2...=g  =  .> 

■•■I%^  =  SC,^    or    |=C,.+  C„ 

-j-dx  z=  f  Cyxdx  4-  /  Cgfi^a;    or    y  =  -^—  +  C^x  +  Cg,  a  parabola. 
21 


PiRT  II. 

RECTIFICATION    OF   CURVKS.    QUADRATURE   OF 
AREAS.    CUBATURE  OF  VOLUMES. 


CHAPTER   I. 


RECTIFICATION    OF    CURVES. 

92.  To  rectify  a  curve  Is  tO  determine  a  straight  line  whose  length 
shall  be  equivalent  to  that  of  the  curve,  or  simply  to  obtain  an  ex- 
pression for  the  length  of  the  curve,  in  terms  of  the  co-ordinates  of 
its  two  extremities. 

93.  Prop.  To  obtain  a  general  formula  for  the  length  of  the  arc 
of  a  plane  curve,  when  referred  to  rectangular  co-ordinates. 

Let  AB  be  the  proposed  arc,  P  a 
point  in  it,  OX  and  OY  the  co-ordi- 
nate axes. 

Put  OD  =  X,  DP 


!/,  AP  =  s 


Then  since  ds  =  dx 


V 


'-%■ 


we  shall  have  by  integration 


1  j_  ^!^'\h 


/( 


(5),     the  required  formula. 
dy^ 


94.  To  apply  (S)  we  replace  -j-^    by  its  value,  in   terms  of  x, 

deduced  from  the  equation  of  the  curve,  and  then  integrate  between 
the  limits  x  =  OE  and  x  —  OF. 


RECTIFICATION  OF  CURVES.  823 

96.  Again  if  y  be  taken  as  the  Independent  variable,  we  shall  have 


/        do? 
ds  =  dy\/  \  -{-  -— -  .  and  therefore 

5  =  /  / 1  +  3~T/    ^y  •  •  •  ('^1)5  ^  second  formula. 

dx^ 
This  will  be  applied  by  substituting  for  -— -    its  value,  in  terms 

of  y,  derived  from  the  equation  of  the  curve,  and  then  integrating 
between  the  proper  limits. 

EXAMPLES. 

96.  1.  To  find  the  length  of  the  para- 
bolic arc  AB^  included  between  the  ordi- 
nates  b^  and  h^. 

The  equation  of  the  curve  is  y^  =  2px. 

dx      y 

.'  ,   -J-  =z—, 

dy      p 
which  substituted  in  [S-^)  gives 

But  by  formula  {B\      %  ^  '^-^  ^ 


Y 

f 

"S D ^=- 


/b'  +  y^)^dy  =  \{p''-\-  y2)^y  -f  \p''f{v'  +  y^)  ^ dy 
To  integrate  the  last  term,  put  (/>2  -f.  y^y  =  z  -\-  y, 
r,p^-\-y'^  =  z'^^zy^y\     y  =tjZ^,     dy  =:  ^'^ 

and 


(1). 


2z 


2^2 


db. 


(^.  ,,,.)*., +^^-^^_i>^  +  ^^ 


2^ 


2z 


■    ■■■fiv^^f)-'^y=-f^.■'^^^=-f^=-^o,.^c. 


324 


INTEGRAL    CALCULUS. 


And  by  substitution  in  (1), 

/0>»  +  y')^dy  =  \{f  +  y^.y  -  \p'.-\og  [(p^  +  y^*-  y-\  +  C,. 
•  •  •  *  =  ^1^-^  - 1^ .  log  [{P^  +  y^)*  -  y]  +  C,. 

-   To  determine  the  value  of  C^,  put  y  =  h^  and  s  =  0,  since  the 
arc  is  supposed  to  commence  at  the  point  A. 

Thus     0  =.  (^'  +  V)^._lyi„g[(^.  +  j^.)i_  j_]  +  (7^. 


i, 


substitution 
when  y  =  ^2 

2i?  ^^-/^  2  /     2J_A2N*        A 

If  the  arc  be  reckoned  from  the  vertex   0,  the  ordinate  h-^  =  0, 

.  _  (i>^  -f-   V-)^6,         1        ,    ^  (/>^  +  V)*-  ^2 

*   •      ■"  2/?  2^     "  j9 

~  2p  "^2^    ^  i9 


2.  The  cycloid      y  =  y^2r^—  x^  ■{■  r .  versin""^ 


Here 


dy        l^\ 


2r  —  X 


and 


dy"^  2r  —  r     2r 


RECTIFICATION  OF  CURVES.  326 

Hence  by  substitution  in  formula  (S). 

4-  =  J   -^  dx  =  2 -v/S^  +  C. 


But  when    a;  =  0,    s  =  0,    .  * .    (7=0,     and  hence     s  =  2y^^rx, 

or,     the  cycloidal  arc  OP  =  2  chord  01.  of  the  generating  circle. 

When         X  =  2r,     s  =  arc  OF  A  =  2  diameter  00. 

. ' .    arc  AOB  of  the  entire  cycloid  =  4  diameters  of  the  generating 
circle. 

3.  The  circle  y*  _  ^2  _  ^^.2^ 

d^  __x     j,^^_j,^_^. 

.  • .       s  z=  I  ~dx=:ir  I  — —  =  r .  sin~i — |-  €. 

^  y  ^  -y/^-x^  f 

This  result  involves  a  circular  arc,  the  very  quantity  we  wish  to 
determine,  and  is  therefore  inapplicable. 

To  obtain  an  approximate  result,  expand  the  differential  coefficieni 

(r^  —  x"^)      and  integrate:  thus 

r/l   ,   1     ^2      1.3    a:*      1.3.5    x^  ^  .      \. 

Vx  ,      \       a:3  ,      1.3       a;5  ,      1.3.5       a:\   ,     1       ,, 

=  4r+2:3--^+2:4r5-7H+2:476:7*7^+H  +  ^- 

But  if   5  =:  0   when   a;  =  0,   then    (7  =  0,   and   .  • ,  when  a;  =  r, 

the  value  of  the  arc  APB  of  the  quadrant. 

And.f    r=l,    .^-.=:l-f_  +  ^-^  +  ^-__4.&o. 


326  INTEGRAL  CALCULUS. 

4.  The  ellipse  a^y^  -f-  ^^-^^  =  a^6^. 

^  dx'~      "^  ttV^  ~        a?{aW  —  b^x^)        ~       ^2  _  ^2^      » 

a2_62 

^^ ^^ 

,     ,    dy'"  a?  a?'  —  c2a;2 

01     1  +  -—.  = — " =  — -,  where  e  is  the  eccen- 

ax^  a?  —  x^  a?'  —  x^ 

tric^ty. 

/.  (a2  _  ,2^2)4               ^  (1  _  ,2,^^2)4  ^ 

.     *  =  /  -^^ ~  dx=  j -~^dx,  by  making  -  =  x^, 

{a^-x'^f  {l-x^^Y  "* 

This  expression  has  already  been  integrated  approximately. 

5.  To  determine  what  curves  of  the  parabolic  class  are  rectifiable. 
The  equation  of  this  class  of  curves  is  2/"  =  aa;*",  in  which  n  and 

m  are  positive  integers. 


dy      m    -    — I 


1^ 

(m  —  n\ 
n     ) 


*  1 

and  this  can  be  rationalized,  when  — -; =z  r,  an  integer,  that  is, 

when    —  = (Art.  41 ). 

n  2r 

Hence,  if  one  exponent,  n,  be  even,  and  the  other,  m,  greater  by 
unity,  the  curve  will  be  rectifiable ;  that  is,  an  exact  expression  for 
the  length  of  the  curve  can  be  obtained  in  terms  of  the  co-ordinates 
of  its  extremities. 

The  term  rectifiable  is  sometimes  restricted  to  those  curves  whose 
lengths  can  be  expressed  algebraically,  or  without  employing  tran- 
scendental quantities ;  and  with  this  restriction,  the  value  of  r  must 
be  positive,  otherwise  s  would  be  transcendental. 

Now   applying   the   other   condition   of   integrability,  we   hav» 

h  -  =  r,  an  mteger,  whence  — =  — - — • 

w  —  n\      2        '  ^    ■  m  2r 


(m  —  n\ 


RECTIFICATION  OF  CUIU^ES.  827 

Hence,  if  one  of  the  exponents  be  an  even  integer,,  and  the  other 
less  by  unity,  the  curve  will  be  rectifiable. 

Combining  the  two  results,  we  find  it  simply  necessary  that  m 
and  n  should  differ  by  unity. 

97.  Prop.  To  obtain  a  formula  for  the  rectification  of  polar 
curves. 

Here  we  have  to  express  s  in  terms  of  r  or  &,  and  for  this  pur- 
pose we  must  transform  the  formula  [>S'],  by  means  of  the  relations 

u^         ux         dij 

^2  =  ^+i-2"--(l)-    ^  =  ^-cos^...(2).     y  =  /-sind...(3), 

the  quantity  ^  being  taken  as  the  independent  variable. 
Then  (2)  and  (3)  give 

dx  dr  dy  dr 

-jT  =  —  r  sin  ^  -f  cos  6—^     and     -rr  =  r  cos  &  +  sin  d  -— • 
dd  d6  d&  ^  dA 


^d&^ 


ds^  dr  d^'"^ 

.  • .   -7-=!  r^sin^^  —  2r  sin  ^  cos  ^  — ■  +  cos^^  -— 

d^^  d&  d&^ 

dr  d?'^ 

-f  r^cos^^  +  2r  sin  &  cos  ^  -rr  -f  s'mH  -— 
dd  dd^  , 

—=/[-+!:> (n 

1.  The  logarithmic  spiral  r  =  a^,  between  the  limits  r  =  r^  and 

-—  =  log a.a^  =  — ,  where  m  is  the  modulus. 
dd         -^  m  ' 

.  • .  d&  =  —  dr  =  —  dr,  and  by  substitution  in  {T), 
aO  r 

. . .    s  =  f(r^  +  ^\  ^dr  =  (m2  +  l)^fdr  =  {m^  +  l)i  +  C 

But     5  =  0,     when     r  =  r^,     .',   C  =  —{m^  +  l)\, 
.•.  s  =  (1  -4-  m2)*(r— rj),  and  when    r  =  r^,    s={l-{-m'^y{r^—r^. 


828  INTEGRAL  CALCULUS. 

2.   The   spiral   of  Archimedes   r  =  a^,   from    the   pole   to   the 
point   r  =  Ty 

dd  a  a*f  ^  ' 

This  expression  is  entirely  similar  to  that  integrated  in  rectifying 
the  parabola. 

. ..  .  =  !i(^l±Ij!)*+  la  log  '•-  +  (''^  +  '-'°)*  . 

3.  The  lemniscata      r"^  =  a^cos  2^,  ^ — ^ 


*=~'-^'  „.A_ii\i  .„4_,,^i 


«^(l-^)*  («*--*)' 


(a*  —  r*) 


1.8.5    r'2  .     "1  . 


+  3 

which,  integrated   from  r  =  a  to  r  =  0,  gives  for  the  arc  BIA  or 
one-fourth  of  the  entire  length  of  the  curve. 

*  =  4^+2:5  +  2:4:^+2:47on3'H 

98.  When  the  curve  is  characterized  by  a  relation  between  the 
radius  vector  r  and  the  perpendicular  p  upon  the  tangent.  To 
obtahi  a  formula  for  the  rectification  in  this  case,  we  assume  the 
value  of  the  perpendicular  found  in  the  Differen.  Calculus, p.  154 viz.: 


\^. 


)v 


^t 


QUADRATURE  OF  PLANE  AREAS. 


cfe2 


c/r2 


J92      ■     (i&2  p^  y.2^,.2  — p2^  y2  — ^2 


-  =  /- 


rdr 


(CT),  the  required  formula. 


£!».  The  involute  of  the  circle  from  ^  =  0  to 
Here  the  equation  of  the  curve  is  r^  =  a^  -f  p^. 

*f     a         2a  2a 

But  when  ^  =  0,  s  =  0.     .'.  C  =  —  -—',  and 


829 


when 


V  =  2'ff'a, 


5  =  2'7r%. 


CHAPTER    II, 


QUADRATURE  OF  PLANE  AREAS. 

99.  The  quadrature  of  a  plane  curve  is  the  determination  of  a 
square  equal  in  area  to  the  space  bounded  in  part  or  entirely  by  that 
curve.  The  problem  is  regarded  as  resolved  when  an  expression  for 
the  area  in  terms  of  Itnown  quantities  has  been  obtained,  the  number 
of  terms  being  limited. 

100.  Prop.  To  obtain  a  general  formula  for  the  value" of  the  plane 
area  ABCD^  included  between  the  curve  BC^  the  axis  OX,  and  the 
two  parallel  ordinates  AD  and  BC^  the  curve  being  referred  to 
rectangular  co-ordinates. 

Put  OE=x,  EP=y,  EF=h,  FP^^y^, 
and  the  area  AEPDz=:A. 

Then  when  x  receives  an  increment  A, 
the  area  takes  a  corresponding  increment 
EPP^Fy  intermediate  in  value  between  the 
rectangle  FP  and  the  rectangle  FS. 


A    EF 


330 


INTEGRAL  CALCULUS. 


But 


C3FS 

niFF 


ViXh 
y  Xh 


,   dy 


h_      d^i 
1    '^  dx^ 


1.2 


+  &C. 


^2 


dx  y       dx^   1.2.y 


+  &c.  =  1,  when  A  =  0. 


Hence  at   the  limit,  when  h  is  taken   indefinitely  small,  the  area 
EPP-^F^  which  is  always  intermediate  in  value  between  FP  and  FS^ 
must  become  equal  to  each  of  these  rectangles,  or  equal  to  y  X  h. 
.  • .  dA  =  ydx,  and  consequently 
A  =  fydx (  ^)?  the  required  formula. 

101.  If  the  area  were  included  between 
two  curves  2)0  and  D^Ci^we  should  find  by 
a  similar  course  of  reasoning 

A  =  /{Y-y)dx (V,), 

in  which  Y  and  y  denote  the  ordinates 
FP  and  FPi,  corresponding  to  the  same 
abscissa   OF. 

102.  To  apply  ( F)  or  ( V{),  we  eliminate  y,  or  y  and  y,  by 
employing  the  equation  of  one  or  both  curves,  and  then  integrate 
between  the  limits  x  =  OA  and  x  z=  OB. 


EXAMPLES. 

103.    1.  The  area  A  BCD,  included   between   the  parabolic  arc 
DC,  the  axis  of  x,  and  two  given  ordinates  AD  and  BC. 

Put    OA=a^,  AD=h^,   OB=az,  BC=:bJ  OE=x,  and  FP=y. 

Then,  from  the  equation  of  the  parabola, 
we  have 


y^  =  2px,     or    y  =  (2p)  .  x  . 
.  • .  And  by  substitution  in  formula  ( F), 


/ 


0    A 


A=f{2p)^.xhc  =  ^^(2p)^.x^+  C=~(2px)\x-\-  C  =  r,^y+  C. 


i 


E  B 

2 
3 


QUADRATURE  OF  PLANE  AREAS.  331 

2 

But        A=  0,  when  x  =z  a^  and  y  —  b^,  .  • .  C  =  —  -  a^b-^^ 

o 

2 

.  • ,  A  =  -  {xy  —  a-Jb^  =  ADPE',  and  when  a:  =  ag  and  y  =  b^ 
o 

A  =  ^{aj>^-a,b^)  =  ADCB. 

Cor.  If  the  area  OB  CB  of  the  semi-parabola  were  required,  we 
should  have 

2  2 

aj  =  0,  ^1  =  0,     and     .' .  A  =- aji^  =  o  circumscribing  □  ; 

o  o 

and  for  the  entire  area  of  the  parabola 

4  2  2 

2A  =~  aj}^  =  -  ttg .  2^2  =  Q  circumscribing  □. 

2.  The   circle    y"^  =  r"^  —  a;^,    or    its    seg- 
ment ACD. 

Here     ^  =  fydx  =  /(r2  -  x'^f  dx, 

or  by  employing  formula  (^), 

Suppose  the  area  to  be  reckoned  from  A. 

area  ^  =  0     when     x  =  OA  ='  —  r, 

.  • .  (7  =  i r2  cos-i(- 1)  =  i -jrrz. 

.  • .  A=i-  -rrr^  -f  -  a;(r2  —  aj^r  —  -  r^  cos-^  — 
2  2  2  r 

And  when  x  =  -\-  r,  A  =  -  itr"^  =z  area  of  semicircle  AUB, 

.  • .  area  of  entire  circle     AEBD  =  itr^. 


S32  INTEGRAL    CALCULUS. 

T^  find  the  area  of  the  aegment  A  CD,  make  x—OG=—a^  then 

.  • .  segment  CADC  =r.  AC  -a.CG. 
3.  The  elliptic  segment         A  C^D^ 
Here  the  equation  of  the  curve  is 

y  =  ^  («2  -  0:2)4 
,* .  A  z=  fydx  =  - / (a^  —  x^)    dx. 


, ' ,  2A  =  segment  A  C^D^  =z  -  .  segment  ACD  of  a  circle  described 
on  AB. 

Hence  the  area  of  the  entire  ellipse  =  --area  circle  =  -"!fa^='!t'ab 

^  a  a 

1  X 

4.  The  cycloid         y  =  (2rx  —  x^)  -\-  r  .  versin-^- 

Put     OD  =  X,     DP  =  y. 
Then  the  area   OPD  =  fydx.  JV^^R/^' 


But  since  y  is  a  transcendental  / 
function  of  a;,  it  will  be  preferable  A 
to  integrate  this  expression  by  parts.     Thus 


A  =  fydx  =  xy  —  Jxdy. 


QUADRATURE   OF  PLANE   AREAS. 
But  from  the  equation  of  the  curve  we  have 


dx 


=xA 


or     dy 


-4- 


2r  —  a; 


dx. 


X  \  X 

.' .  A  =z  xy  —  f  '^^rx  —  x^ .  dx. 

Now  /  -y/2ra;  —  x'^dx  =  fy-^dx  where  y-^  is  the  ordinate  DP-^  of 
the  generating  circle,  corresponding  to  the  abscissa  OD  =  x^ 

or  f  ^'2rx  —  x'^dx  =  area  OP^D. 

.  ' .  area  OPD  —  xy  —  area  OP-J)^    and  when    x  —  OC  z=.^r, 
area  semi-cycloid  OAC  —  OC  X  CA  —  area  semicircle  OP-fi 

1  3 

Z  lit 

. ' .  area  entire  cycloid  =  Sirr^  =  3  area  generating  circle. 

104.  Prop.  To  determine  a  general  formula  for  the  quadrature  of 
polar  curves,  their  equation  having  the  form  r  =  Fd, 

Let  QX  be  the  fixed  axis,  QP  the 
radius  vector,  forming  with  QX  an 
angle  measured  by  the  arc  6  described 
with  radius  equal  to  unity. 

Let  6  take  the  increment  t,  convert- 
ing r  into  rj  =  F(d  -\-  <),  and  adding 
the  sector  QPP^  to  the  area  QIP=A, 
previously  swept  over  by  the  radius  vector, 
but  <  QPiO.     Also  the  ratio 


Now  QPP^  >  QPK, 


qpK 


-r^  X  r^t 


r  X  rt 


,     ,  dr   t    ,  dh'     <2     .    ,     ,, 


=  1  +  2*. i  +  2  — 


=  I     when     t  =  0. 


3^4  INTEGRAL  CALCULUS. 

Hence  at  the  limit,  when  t  is  replaced  by  c?^,  and  QPP-^  becomes 
dA,  the  value  of  QPP^  will  be  equal  to  QKP  or  QP^O.     Thus  we 

shall  have  dA  =-r^  .d&. 

.  • .  A  =  -  fr^d&  .  .  .  .  ( Fg),  the  required  formula. 

1.  The  spiral  of  Archimedes  r  =.  a^> 

A  =  ^  fr^d&=l  a^f&m  =  i  a^^  +  (7  =  i  —  +  (7. 

U  A  =  0  when  r  =  7%,  then   C  =  —  -  — • 

6  a 

For  the  area  of  one  convolution  estimated  from  the  pole,  we  hav« 
the  limits  rj  =  0  and  ?2  =  2'n'a. 

o 

2.  The  logarithmic  spiral  from  r  =  r^  to  r  =  r2. 

Here        r  =  a  .     .'.  dr  =  log  a .  a  .  c?^     and     d&  =  . •  — • 

log  a     r 

.♦.     A=:lfrW=-—frdr  =  j-^r^-^a 
2  2  log  a  4  log  a 

=  T  ^(^2^  —  ^1^)?  between  the  limits  r^  and  rg:  the  quantity 

m  denoting  the  modulus. 

3.  The  hyperbolic  spiral  from  r  =  r,  to  r  =  rj. 

IT  «        J  <*^^        7.  ^^     T  a  , 

Here  r  =  - )      ar  = — j      d^  = ofr  = -dr. 

d  ^2  >/  r- 

.  ♦.     A  =  —  -fadr  =z  —  -ar  -\'  C  =  ^       T         between  the  lim- 


Its  rj  and  r^. 


QUADKATURE  OF  PLANE  AREAS.  335 

4.  The  lemniscata  r^  ==  a^cos  2  4. 

A  =  \:SrH^  =\  a^'f  cos  2m,z=zla^sm  2&  +  C. 
,«■ 

Put  A  =  0  when  ^  =  0 ;  jthen   (7  =  0,  and  ^  =  -  a^sin  2^, 

4 

which  gives,  when     r  =  0,     or     6  = -<ji^,*    A  = -a\ 

. ' .     Entire  area  =  a"^  =  square  described  on  semi-axis. 

105.  Prop.  To  find  a  formula  for  the  quadrature  of  a  plane  curve, 
when  its  equation  is  given  by  a  relation  between  the  radius  vector, 
and  the  perpendicular  upon  the  tangent. 

Since    d^  =  -^'—,.        A  =  \frm=\f-Jr^^ 

1.  The  involute  of  the  circle 

(r^—p^y 

and  this,  between  the  limits  ^  =  0,  and  p  =  2'jra,  within  which  tlie 
entire  circumference  is  unwound,  gives 

A=^  'j(^a\ 

Cor.  The  area  included  between  the  involute  ABS,  the  circle, 
and  the  tangent  AS,  is  equivalent  to  that  swept  over  by  the  radius 

sector,  and  therefore  equal  to  -^« 

c^ir^ ct^i 

2.  The  epicycloid  p^  =     ^^  _    ^\  where   c  =  a  +  26,  a  and  b 

being  the  radii  of  the  fixed  and  generating  circles. 


836 


INTEGRAL    CALCULUS. 


Put        (r-2  —  a^y  =Z,       .  •  .    ^2  _  2;2  _|_  ^2^       ,.^^  _  ^^g^ 
(C2  -  r2j4  ^  (c2  _  a2  -  22)* 

.  • .     ^  =  i  f  /(c2  -  a2  _  z'^f^z'ds 


2;(c2  —  0,2  _  ^2)^c  ^^2  ^  (f2y    /* 


dz 


Aa 


4a 


(C2  _  a2  -  22) 


2^J 


2(c2  _  a2  -  22)*c   .    (c2  -  a?)c    .     ,  r/-2  -  a^^       ^ 
4a  4a  Lc^  —  a^  J 

This,  between  the  limits  r=a  and  r=c, 
gives 

=  ^(a2  +  3a6  +  2b^)  =  0/PFO. 

But  OIL  =  -  t(ah, 

1  62^ 

.  • .   IPVL  =  ^  epicycloid  =  —  (3a  +  26), 

62-r 

and,     IVIyLI  =  — (3a  +  26),  the  entire  epicycloid. 
a 

If   6  =  -  a,  then  epicycloid  =  4'n'62  =  -th^  :=:  area  fixed  circle. 


If   b  =  a,     then  epicycloid  =  5'xb^  =  5'7ra2  =  5  area  fixed  circle^ 


CHAPTER    III. 


QUADRATURE    OF    CURVED    SURFACES. 


106.  Prop.  To  obtain  a  general  formula  for  the  quadrature  of  a 
surface  of  revolution. 

Let  AB  be  the  arc  of  a  plane  curve 
which  revolves  about  the  axis  OX,  P  and 
Pj  points  taken  on  the  curve  so  near  to 
each  other  that  the  arc  PPj  may  present 
its  concavity  to  OX  at  every  point. 

Put      ODz=zx,     DP=.y,     DD^=:h,     A^j  -  y„     JP=«. 

The  surface  generated  by  the  arc  PPi,  is  intermediate  in  magni- 
tude between  those  generated  by  the  chord  PPj,  and  the  broken 
line  PTPy     Denoting  these  surfaces  by  C  and  B^  we  have 

1 


iPD  4-  TD^)2ifPT^-  (TDi'  -  P^D^^)t( 


{PJ)  +  P^D^)PP,.2if 


-  (^-P^  +  VT)PT-^  (^Pi^i  +  PiT)P^T 
~  {2PD  +  VP^)PP^ 

Dividing  numerator  and  denominator  by  A,  and  then  passing  to 
22 


338  '  INTEGRAL    CALCULUS. 

the  limit,  we  obtain  — -  =  1.     And  hence  the  limit  to  the  value  of 
C 

the  surface  (7,  generated  by  the  chord,  will  be  a  proper  expression 

for  the  elementary  surface  generated  by  the  arc  PPj,  when  that  arc 

becomes  indefinitely  small. 

But  at  the  limit,  when     h  =  dx^     C  =  2'jryll  -f  y^l  «^. 

Hence  we  have  for  the  differential  of  the  surface, 

dA  =  2^y(l  +  g) V     and     .'.  A=  2^/J(l  +  g) V^, ..{W)^ 

or,  A  =  2'jrfi/ds (  W{). 

107.  To  apply  (W),  we  eliminate,  by  means  of  the  equation  of 

the  generating  curve,-  y.  and  -j-,   and  then  integrate   between   the 

given  limits.     Similarly,  we  apply  (  Wj)  by  expressing  y  in  'terms 
of  s,  or  ds  in  terms  of  y  and  dy. 

EXAMPLES. 

108.  1.  The  surface  of  the  sphere.  Y 

Here  the  generating  curve  is  a  circle  whose  equa- 
tion is 


y'  = 

r2  -  x\ 

X 

l+^^^ 
'^dx^- 

-X 

+  x^ 

y' 

r2 

~y' 

A  -r   0    +r    B 


r,A  =  2ifj\^  =  2'xrfdx  z=z  2'rrra  -f-  0, 

Put  J  =  0,    when    xL  -^r;    then     C  =  2"^ A_ 

.*.  A  =z  2'ifr(r  -f-  xj,  which,  when  x  z=  -\- r,  gives  for  the  surface  of 
the  entire  sphere  A  =  Airr^  =  4  great  circles. 

For  the  zone  whose  height  is  ?i  z=  X2  —  Xj,  we  have 

A  =  2'n'r{x^  —  x^)  =  2ifrk. 


Y-  a  ^ 


QUADRATURE   OF  CURVED  SURFACES.  ^^     339 


i 

6 

% 

^ 

1.  The  paraboloid  of  revolution, 

,'.  A  =  2'!ff{y'^  +p^Ydx  =  2^f{2px  +  p^Ydx 

=  ^-  {2px  +^2)*+  (7=-g-  [{2px,-i.p^f-  (2px,-^p^)^]. 
If  the  surface  be  reckoned  from  the  vertex,  we  shall  have  x^  =  0. 

2.  The  surface  generated  by  the  revolution  of  the  Catenary  abouV 
its  axis. 

The  equation  of  the  curve  is     s^  _  ^2  _j_  2ax. 

(x  -\-  a)dx                         f       '                   adx              ddx 
. '.  «*  =     \  and  dyz=z^ds^—dx^  =  — ; —  = 

Now,  applying  formula  (  TF'i),'and  integrating  by    a 
parts,  we  have  *"  '  P' 

A  —  2'fffyds  =  2^{ys  —  fsdy)  =  2'jr(ys  —  afdx) 
=  2if(ys  —  ax)  -f-  0. 
But  when       x  =  0^  y  —  ^  and  s  =  0,  .  • .  (7=0. 


.  • .  ^  =  2'n'(?/y^2  _|_  2aa;  —  ax), 
3.    The   surface  generated   by  the  revolution  of  a  semi-cycloid 
about  its  axis. 

Here         dy  =  \/ •  dx    and    *  =  2y2rx=  ySrx. 

.'.  A  =  2irfyds  =  2ir{ys  —  fsdy)  =  2if(ys  -  /2-v/5r^v/?^^-^.(ir) 
=  2'r(yy^  —  ySr~f^2r  —  x,dx) 

=  2'rfy ^8^4-  y^.  ?  (2r  -  ar)^]  +  G, 

32 
But  when       x  =  %  A  ^0,  r  .  C  —  —  —  f(r\ 

o 


340 


INTEGRAL  CALCULUS. 


2  4      16 

32 

and  when  x  =  2r,  A  =  S'rrh-^ —  rrr^  the  entire  suiface. 

o 

4.  The  surface  generated  by  the  revolution  of  the  cycloid  about 
its  base. 

In  the  formula  A  =  2irfyds,  the  quantity  y  denotes  the  distance 
of  a  point  in  the  revolving  curve  from  the  axis  of  revolution,  and 
must  therefore  be  replaced  in  the  present  instance,  by  2r  —  ar. 

.  ♦ .  A  =  2'r^f(2r-  x)ds  =  2'7r/(2r  —  x)i    '  ^'* 

=  2^  V^(4ra:^  -  I  ^^)  +  0.^ 
o 

But  .4=0,  when  x  =:  0,  y.  (7=0, ,.  •.  ^=2^ ^/27(4ra;*-?a:^); 

o 

and  when  x  =  2r. 

.  • .  A  =:-^  tir^ ;  and  the  entire  surface  2 A  =  — ■  'n'r'^. 
o  o 

109.  Prop.  To  obtain  a  general  formula  for  the  quadrature  of  any 
curved    surface,  whose  equation    is 
referred  to  rectangular  co-ordinates. 

Let  CAPB  be  a  portion  of  the 
surface  included  between  the  planes 
of  xz  and  yz^  and  the  planes  BP^^ 
AP^  drawn  parallel  thereto. 
:  Put  OA^  =  x,  OB^  =  A^P^  =  y, 
P^P  =  0,  A  GBP  =  A,  and  let 
z  =  F{x,y) ...  (1)  be  the  equation  y^ 
of  the  surface. 

Then,  since  the  value  of  ^  will  be  determined  by  the  assumed  values 
of  the  independent  variables  x  and  y,  we  shall  have  A  =  (p(x,7j). 

Now  when  x  receives  an  increment  A^a-^^  =  A,  the  area  A  takes  th« 
increment  AB,  becoming 
A         /     .   J    \        .   ,  dA    h       d^A    h?    ,   cPA       A3       .    . 

1      n    -r^  ,y;  -r  ^^     I  ^  dx''    \.2^  dj^    1.2.3^^^^ 


QUADRATUKE   OF  CURVED   SURFACES.  341 

Similarly,  when  y  alone  takes  an  increment  B-p^  =  k,  A  takes  the 
increment  BG,  becoming 

,      dA    k    ,   d?A    k''       (PA       k^       ,    , 

But,  when  x  and  y  increase  simultaneously,  A  takes  an  increment 
AB  -\-BG-{-  PI,  becoming 

_  dA    h      dA   k      (PA     h?         d-'A    hk      d^A     B 

d^A       h?  d^A     h^k         d'^A      hk^       d^       k^ 

^  "^  *  r."2T3  "^  dx^dy '  172  "^  dxdy^ '  172  "^  'd^ '  17273  "^     ^ 

.-.  FI=A,-A-(A,-A)-{A,-A) 

d^A    hk  .     (^3^     k'^k    .     f/3^4      AF 

+  ZTTJ- '  T—^  +  T-7T  •  t-t:  +  <Sce. 


6^xc/y     1        dx^dy    1.2       rf^t/y2    1.2 

PI      d^A        d^A       h  d^A        k 

lUc  ■"  dxdy  "^  dx-'dy '  \  .2^  dx  .dy''' \7z'^ 

which,  at  the  limit  when  h  =z  0  and  ^  =  0,  reduces  to 
PI        d^A 


J 


(!)• 


Pill       dxdy 

Now  this  quotient,  which   results  from  dividing  the  elementary 
surface  PI  by  its  projection  Pj/j  on  the  plane  of  xy,  is  equal   to 

,  where  v  denotes  the  angle  formed  by  the  tangent  plane  at  the 

point  P  with  the  plane  of  xy. 

But  from  the  theory  of  surfaces  (DifF.  Cal.,  Art.  177),  we  ha\B 

1 


V  c^a:2  "^  dy^ 


d^A 


dz^        dz^ 


dxdy       V  dx"^        dy 

d^A 
Now,  since  the  second  differential  coefficient  -r—T-  is  obtained  by 

dxdy 


342 


INTEGRAL  CALCULUS. 


^=//( 


differ'^ntiating  the  function  A  of  x  and  y,  first  as  though  x  were 
alone  variable,  and  then  as  though  y  only  varied,  we  shall  obtain 

d^A 
the  value  of  A  by  multiplying  the  value  of  -7—7-  by  dxdy,  and  then 

performing  two  successive  integrations  with  respect  to  x  and  y,  the 

order  of  these   integrations   being   immaterial,    since   that   of  the 

differentiations  is  arbitrary. 

This  double  integration  is  indicated  by  the  symbol  //,  and  the 

result  is  called  a  double  integral.     Thus 

dz"^       dz^  \4 
1  -\-  —  +  —^xdxdy  .  .  .  .  (  TFg),    the  required  formula. 

The  limits  of  these  integrations,  in  the  case  represented  in  the 
diagram,  are  y  =  0  and  y  =  OBj  =  b,  x  =1  0  and  x  =  OA^  =  a. 
But  if  the  surface  were  terminated  laterally  by  a  cylinder  (instead 
of  by  planes  parallel  to  xz  and  yz),  the  elements  of  this  cylinder 
being  parallel  to  the  axis  of  z,  and  its  base  in  the  plane  of  xy  repre- 
sented by  the  equation  y^  =fa,  then  the  superior  limit  of  the  first 
integration  would  he  y  =  y-^  =f^^  the  inferior  limit  being  still  zero. 
This  will  be  rendered  plain  by  an  example. 

110.   1.  Required  the  surface  of  the  tri-rectangular  triangle  ABC, 
From   the   equation   of   the    surface  q 


x^-hy^ 

+  Z"^  =  1 

•2,  we 

obtain 

dz 

dx  ~ 

—    and 

z  ■ 

dz 
dy 

-  -I. 

z 

• 

■V- 

dz" 
dx^ 

r 

~  z 

.      A 

rr^.. 

7n. 

.ff 

dxdy 

r/sin-i  (  ^ \dx. 


The  limits  of  this  first  integration  with  respect  to  y  are  y  =  0 


and  y  =  Vr^^-a;^  =  DE. 


QUADRATURE   OF  CURVED  SURFACES. 


348 


But  when  y  =  0, 

and  when    y  =  -y/r^  —  x^, 


V^2 

y 


^r 


=  0, 


1,    and    sin-i(l)  =  ^«'. 


2      -^  2  2 

between  the  limits  x  =  0,  and  x  =  r. 

2.  The  axes  of  two  equal 
circular  semi-cylinders  in- 
tersect at  right  angles,  form- 
ing the  figure  called  the 
groin.  Required  the  entire 
surface  intercepted  upon  the 
two  cylinders. 

Assuming  the  axes  of  the 
cylinders  as  those  of  x  and 

y  respectively,  the  equation  of  the  cylinder  whose  axis  coincides 
with  X  will  be  y^  -\-  z^  =  r^,  and  that  of  the  cylinder  whose  axis 
coincides  with  y  will  be  x^  -{-  z^  =  r"^. 

The  entire  surface  to  be  estimated  is  projected  upon  xy  in  the  rec- 
tangle ABCF,  and  the  triangle  OGF  is  the  projection  of  one-eighth 
of  this  surface.     To  compute  this  portion    to  which  the  equation 

^2  _|_  2;2  _  ^2  applies,  we  have  A  —J  /  I  1  +  t"^  +  'T^\  dxdy,  in 
•.vhicb  the  limits  of  integration  are  y  =0  and  y  =  x,  x  =  0  and  x  =  r. 

dz  X  dz       ^ 


Jut  from  the  equation    x"^  -\-  z^ 


dx 


•  •  •  ^  =//(i + ip'^y  =ffl  '^'^y  =ff 


rdxdy 


ydx 


xdx 


or 


Vr 


V-r 


between  the  given  limits. 


x"        *^  -x/r"  —  X'- 

A  =  —  r-s/i^  —  a;2  -j-  C  =  r"^  between  the  limits  a:  =  0  and  x  —  r, 
,'.  SA  =  8r2,    the  entire  surface  of  the  groin. 


CHAPTER    IV, 


CUBATURE    OF   VOLUMES. 

111.  Prop   To  obtain  a  general  formula  for  the  volume  generated 
by  the  revolution  of  a  plane  figure  about  a  fixed  axis. 

Let  OX,  the  axis  of  a:,  be  the  axis 
of  revolution,  ABCF  the  generating 
area.     Put 


OD=x,  DP=y,  DD^=h,  D,P,=y^, 

and  let  y  =  Fx  be  the  equation  of  the 
bounding  curve  AB. 

The  volume  generated  by  the  revolution  of  the  small  quadrilateral 
DPP^D^  is  intermediate  in  magnitude  between  the  cylinders  gene- 
rated by  the  rectangles  PD^  and  FD^     But 


cylinder  FDj^  _  ity-^h  _  y{* 
cylinder  Pi>i  "  ify'^h  ~  y^ 


dy   h       dhj      h^ 


A2 


^     dx   y^     dx-^   1.2. y 


dy^     h^ 


dx^    y^ 
=  1     when     h  =  0. 
Therefore  at  the  limit  the  volume  generated  by  DPP^D^  —  cyl 
inder  PD^^  or  dV  =z  ify^dx,  and  consequently  V  =  itfy'^dx  . . .  (X) 
the  required  formula. 

To  apply  (X),  we  substitute  for  y^  its  value  in  terms  of  x  derived 
from  the  equation  of  the  bounding  curve  AB.,  and  then  integrate  be 
tween  the  given  limits. 


CUBATURE  OF  VOLUMES.  345 

112.   1.  The  sphere. 

Here  the  equation  of  the   circle  which  bounds   the  generating 
area  is  a*^  -4-  y'^  ==  r^, 

o 
I*ut        V  =  0     when     ar  =  —  r. 


/I  \      2 

then        6'  =  —  <  I  -  r^  —  r^  j  —  -  ■n'r^. 


0      +T       B 


1  2  4 

.-.  V  =  'Ti[r'^x  —  -  a.-3)  -f-  -  <r?-3     and  when     a;  =  -|-  r,  V  —  ~  icr^. 

f      2.  The  ellipsoid  of  revolution,  generated  by  the  revolution  of  the 
semi-ellipse  about  its  greater  axis  2a. 

Here  y''z=z-{a^-x'').    .'.  V  =  -^f(a^-x^)dx=--(a'^x-=;X^)-{'a 

which  gives  between  the  limits  x  =  —  a  and  x  =  -{-  a. 

4  2  2 

F  =  -  -Trb^a  =z  -  {2a .  tb"^)  =  -  circumscribing  cylinder. 

^    3.  The  paraboloid  of  revolution 

y2  _  2px.      V  =  2'rt'p  fxdx  =  'rrpx^  +  C. 
If      V  =  0  when    a;  =  0  ;    then    C7  =  0   and    V=  tipx^ ; 
which  becomes,  when  a:  =  iCj  and  y  =  y^, 

F  =  -TTjoar^^  —     x^.  t(y^  =  -  circumscribing  cylinder. 

4.  The  parabolic  spindle  generated  by  the  revolution  of  the  para« 
bolic  area  A  QB  about  the  double  ordinate  A£. 

Put  OQ  =  a,   OA  =zb,   01)  =  X,  DP  =  y.     Then  QC  =  a  -  y, 

rt      ic2\2         ^ 
x^=.  2p{a—y)  and  V='^J  \^~  K')  ^^—  T~J^^^P'^~'  4cfpa;2  4-a;*)c?ar. 


Z46 


IXTEGRAL  CALCULUS. 


But  if  F  =  0  when  x  =  0,  then  (7=0;    and  when  x  =  OA  =  6, 


or  since 


2   .   1 


O      D 


V  =  ifa^  (l  -  3  +  5)  =  15  '^<*^*  =  volume  ^^0. 

1  (\ 

.  • .  volume  AQB  =  — - ira^. 
15 

5.  The  volume  generated  by  the  revolution  of  the  cycloid  about 

its  base. 

Put     0V=  2r,     ODz=:x,     DF  =  y,     /F  =  0  =  2r  —  y. 

Then  from  the  equation  of  the  cycloid, 

dx       /2r  —  z\^ 
dz 

V 

2^-n4  ^--rr^T---^  p 


But  by  formula  (.4),  T4|«/I1 

fyk^r-y)~^dy=-  \y\2r-yf-{-\rJ\}{2r-y)^dy 

fy\2r-y)~^dy=--y^{2r-y)^^  lrfy\2r-y)~^dy 

Also       fy~^{2r  -  y)~^dy  =  f -—1=-  =  versin-i  h 

''y/'Zry-y-  ^ 

.',  F='rr(2r-y)*Qy*  +  ^A+^A'^)-^'^^-'.versin-i^  f  C. 


CUBATURE  OF  VOLUMES.  347 

5 

Put  V  =  0  and  y  z=  2r.     Then   C  =  -rV2,  and  when  y  =  0. 

5 

F  =  -  /•3'ff2  =  volume  B  VO. 

5  5 

,  • .  volume  £VA  =  Sr^-n'^  =  ^'n'r-itr^  =  -  circumscribing  cylinder. 

2  8 

C.  The  volume  generated  by  the  revolution  of  the  cycloid  about 
its  axis.     (See  last  Fig.). 

Put  VI  =  X,  IP  =  y,  VO  =  2r,  and  to  facilitate  the  integration, 
introduce  the  variable  angle  VCU  =  &. 

Then     x  =  r{l  —  cos  &),  y  =  r(sin  &-{-&),  dx  =  r .  sin  &dd. 
.  • .  r  =  irfy^dx  =  crr^  f{s\n^  +  2^  .  sin2^  +  &^ .  sin 


12  4 

But  /sin3^  .c^^  =  ~~  Q  ^^^^^ .  cosd  —  -  cos  ^  =:  -,  from  ^  =  0  to  ^  =  -r. 
o  So 

2/(3 .  sin24  .dd  =f&.d&  -fd .  cos  2m 

=  -^2_^sin2^+-  /sin  2^ .  d&  by  integrating  by  parts. 

=  -^2_^sin2^— -cos2^  =0*^^  from  ^  z=  0  to  ^  =<r, 
2  2  4.  2 

and  /^2, sm&.d^  =  -~  &^ cos ^  +  2 /^  . cos ^c/J 

=  _  ^2  cos  ^  +  2^  sin  ^  —  2 /sin  ^fl?d 

=  _  ^2  cos  ^  -f  2&  sin  ^  +  2  cos  ^ 

=  -^2  _  4^  from  a  =:  0  to  ^  =  <r. 


-■^-2  —  -J. 


113.  Prop.  To  obtain  a  general  formula  for  the  volume  of  all 
solids  which  are  symmetrical  with  respect  to  an  axis. 

Such  solids  may  be  generated  by  the  motion  of  a  plane  figure,  as 
ABCD,  of  variable  dimensions,  and  of  any  form,  whose  centre  G 
remains  upon  the  axis  OX,  its  plane  being  always  perpendicular  to 
OX,  and  its  variable  area  X  being  a  function  of  ar,  its  distance  from 
the  origin. 


348 


INTEGRAL   CALCULUS. 


By  a  method  entirely  similar  to 
that  applied  to  solids  of  revolution, 
we  may  show  that  dV  :=  Xdx, 

and      .-.  V=fXdx (Zi), 

the  required  formula. 

To  apply  (JT,)  we  must  express  the 
value  of  the  area  X  in  terms  of  x,  and  then  integrate  between  the 
proper  limits. 

Cor.  The  same  formula  is  applicable  to  any  solid  generated  by 
the  motion^of  a  section  of  variable  dimensions  parallel  to  a  given 
plane,  when  the  area  of  the  section  can  be  expressed  in  functions  of 
its  distance  from  the  fixed  plane. 

114.  1.  The  ellipsoid  with  three  unequal  axes. 

x^        Ip"        z^ 
Here  we  have  -—  -f  ■— -  4-  -::  =  1 , 

a^        b^        c^ 

or  b'^c'^x'^  +  a^c2^2  _|_  ^2^2-2  _  a^b^c^. 

Make  CC-^  =  x,  and  put  successively 
y  =  0    and    z  =  0. 

Then  when 

c 


y  =  0,   z  = 
and  when 


2  _  ,,.2 


A^i; 


z  =  0,  y  =  --y/^-  x^  =  D,C\. 


J] 

3 

{^. 

i, 

^ 

^ 

c 

[^ 

^'j 

V 

y 

ll 

oi 

^ 

«jr5c 


.  • .  area  B,D,F,E,  =  X  =  —  (a'-  -  x^)  ; 
and  this  value  substituted  in  {X{)  gives 

V  =  ^/(a2  -  x'^)  dx^~  {a'x  -  ^a;3)  -f  C. 
Put  F  =  0  when  a;  =  --  a  ;  then   C  =  —^  •  ^  «^  and  when  ar=  -\-a 


V  =  -itbca  z=z  entire  ellipsoid, =  -  circumscribing  cylinder. 


CUBATURE  OF  VOLUMES. 
2.  The  elliptical  paraboloid  cz^  +  ^"^  =  «^^» 


349 


Put  successively 


then 
Then 


CB 


y  =  0  and  «  =  0 ; 
""^^     and     CD  =  a 


x\i 


y/bc 


X  =  —-=.'     And 
-y/bc 

r      ,       1  tid^x^ 
I  x.dx  =  -  —-= 


+  0, 


If     r=0    when    X  =  0,    then    C  =  0,    and 

•.  • .  When  X  =  0^  =  arj    ^  =  «  ~~F==  =  o '^i^i  —  o  circumscribing 
cylinder. 

3.  The  groin  or  solid  formed  by  the  intersection  of  two  cylinders 
whose  axes  are  perpendicular  to  each  other. 

1st.  Let  the  bases  of  the  cylinders  be 
equal  senn-eircles. 

Then  the  generating  section  A-^BiC^D^ 
will  be  a  square. 

Put    OG^GE=  EA  =  r,  OG^  =  x, 
G,E,^y=E,A,, 

Then    A^B^C^D^z=z  Aif,    and  from   the 
equation  of  the  circle  EOF^ 
y2  _  2rx  —  x^.  .  • .  V~  fXdx  =  f{Srx-4x^)dx  =  4rx^  _  | a:^  -f  (7. 

But    V  =  0  when  x  z=  0,  .  * .  (7=0,  and  when  x  =  r, 

8  2  2 

pr_      r^=  -  r  .2r  .2rz=  -    circumscribing  parallel opipedon. 

2d.  Let  the  bases  be  unequal  parabolas. 
Then  the  generating  section  will  be  a  rectangle. 
Put  OG  =  a,  GE=b,  EA=b^,  OG^z=x,  G^E^  =  y,  ^,^,  =  y, 


850 


INTEGRAL  CALCULUS. 


Then 


2pa;,  y^  —  ^p^x.  .  • .  X  =  2y .  2yi  =  ^x^/pp^. 


V ."  fXdx  —  ^y/vPi  fxdx  =  Ax'^^pp^  =  2x .  yy^,  and  when  ar  =  a. 

V z=z  2abb^  —  -a .  26 .  26^  =  -  circumscribing  parallelopipedon. 

4.  The  Conoid,  with  a  circular  base. 

Put     DA  =  a,  DE  -  2r,    DG  =  x,   GI  =  y. 

Then  the  generating  triangle  IFH  z=:  X  z=  ay 


=:  a-\J'lrx  —  x^. 
.  ' .    V  :=  fXdx  ~  a  f^2rx  —  x'^.dx 

—  a  .  segment  BGH, 

and  when      x  ~  2r,    F  =  a  •  (semi-circle  DHE). 

or  volume  conoid  =  -•  volume  circumscribing  cylinder. 

Cor.  A  similar  result  will  be  obtained  if  we  suppose  the  base 
to  have  any  other  form,  the  generating  triangle  being  still  perpen- 
dicular  to  the  base. 

115.  Prop.  To  obtain  a  general  formula  for  the  volume  of  asolicl 
bounded  by  any  curved  surface,  whose  equation  is  referred  to 
rectangular  co-ordinates. 

First  suppose  the  volume  bounded  by  the  co-ordinate  planes  of 
xy^  xz^  and  yz^  by  planes  parallel  to  xz  and  yz^  respectively,  and  by 
the  curved  surface  (7a/6,  whose  equation  is 
^1  =  F{x,y), 

Put   OA^  =  X,  OB^  =  A^P^  z=  y, 

PiPi  =  ^,  A^  =  ^v 
A^a^  =  dx,  P^  G-i  =  dy^  PiP  =  ^^^' 
Let  the  volume  be  intersected  by 
planes  AG^  and  a/j,  parallel  to  yz, 
and  including  between  them  the 
lamina  or  slice  Ail:  let  this  la- 
mina be  cut  by  planes  6/j  BD^^  &c., 


CUBATURE  OF  VOLUMES.  351 

dividing  /t  into  pri»ms  such  as  P/j,  &;c. ;  and,  finally,  let  each  prisna 
be  subdivided  into  elementary  parallelopipedons,  such  as  z-^d  by 
planes  parallel  to  ary,  the  successive  planes  being  at  distances  from 
each  other  denoted  by  dx,  dy^  and  dz^  respectively.  Then  the 
Volume  of  one  of  these  elementary  parallelopipedons  will  be 
oxpr.;ssed  by  dxdydz\  and  if  this  be  integrated  with  respect  to  2, 
regarding  x  and  y  as  constant  between  the  limits  2;  =  0  and 
z  ■=:  z^rzz  P^P  =  F(x,y)^  the  result  obtained  will  represent  the  sum 
of  all  the  parallelopipedons  contained  in  the  prism  P/j.  A  second 
integration,  with  respect  to  y,  between  the  limits  y  =  0  and  y=A-^G-^, 
will  give  the  sum  of  the  prisms  contained  in  the  lamina  AI-^ ;  and  a 
third  integration,  with  respect  to  .r,  between  the  limits  x  =  0  and 
X  =z  Oa^,  will  give  the  sum  of  the  laminae,  which  constitute/the  entire 
volume.  /  ^  /  . 

Henc"  the  required  formula  is  »       J^     ^^^/  ^  ^t^ 

V=fffdxdydz (1).    {     ^         ''/•  •' 

The  symbol  ///denotes  three  successive  integrations^wiuriie^aeol       "  /  y, 
to  the  variables  ar,  y,  and  z^  and  the  result  is  called  the  triple  mtegrat  J ^ 
of  dxdydz.  ^  *  / 

Cor.  If  the  volume  were  bounded  on  every  side  by  the  curved 
surface,  the  same  formula  (1)  would  apply,  but  the  limits  of  inte- 
gration would  be  different,  those  of  the  first  integration  being 
z  =z  z-^^  and  z  =  z^  where  z^  and  z^,  are  the  two  extreme  values 
of  z  corresponding  to  the  same  values  of  x  and  y,  and  derived  from 
the  equation  of  the  surface ;  those  of  the  second  integration  being 
y  z=zy^  and  y  =  7/3,  the  extreme  values  of  y  corresponding  to  the 
same  value  of  x^  and  derived  from  the  equation  of  the  section  per- 
pendicular to  0X\  and,  finally,  those  of  the  third  integration  being 
t^  and  0^2,  the  extreme  values  of  x. 

116.   1.  The  tri-rectangular  spherical  sector. 

Here  the  limits  of  the  integration  are  ^=0  and  P^P—  -y/r"^ — x^  — -y*, 
y=0  and  y=.ByEz=L ^r^—x"^,  ar=0,  and  x=zOA=r. 


352  INTEGRAL  CALCULUS, 

.  • .    V  z=  fffdxdydz  =  ffzdxdy 


—  ff^r'^  _  a,.2  _  y2 ,  dxdy.       But 
/-v/r2— :r2— y2.  dy  —  -y(r'^—x'^—y'^) 


+J('-'-^')/7S 


G?y 


?/(/2-a;2  — ?/2) 


l//-2— if2— y2 


4.1(^2-^2)  sin-i--^=l 

'*  '  ^   2_,r2        4 


v- 


=:-';r(r2— a:2)  between  the  liiriits  given. 


V—x 


between  the  limits. 

2.  The  volume  cut  from  a  paraboloid  of  revolution,  4;he  equation 
of  whose  generating  curve  is  y^  =  2px^  by  a  right  cyliii^ '•■  with  a 
circular  base,  its  axis  passing  through  the  focus,  and  the  diameter  of 
its  base  being  equal  to  p. 

The  equation  of  the  paraboloid  being  y2  _|_  2;2  —  '2/jx,  and  that  of 
the  cylinder  y^  z=z  px  —  x^,  the  limits  of  integration  in  th-  present 
case  will  be 

2=4-  ■x/'Zpx  —  y2     and     z  =  —  y^2/jx  —  y^, 


y  =  4-  ypx  —  ic2     and     y  =  —  ypx  —  a;2, 

X  =  0     and    x  =  p. 

\  V=fffdxdydz=ffzdxdy=ff2(2px-y^)^dxdy.    ° 


ut  f(2px^y^)^dy=ly{2px-y^)^+px  fjJL= 
-^  *^y2px—y^ 

1  1  « 

=  o  y(^P^  -  2/^)  +  P^ '  sin-^  -7= 

=  a:y/y2~l-  a:2  4.  2jo.r .  sin-^  \/^ — ~  between  the  limits. 


CUBATURE  OF  VOLUMES. 


858 


dx 


=  —  ^  (j>2  _  a;2)*-|-  2j9^2^sin~^\  7'^: 


—  a; 


2  |. 

o 


2p 


p—x 


+ 


rf 


x^dx 


v//>^~--  x^ 


-  gM/^'^ 


=4+9 


'2\^  I  1    T  •    1  ^ 

^  )  H —  /^   sm-i  - 
2  ^ 


between  the  limits  x  =  0  and  a;  =  j!7. 


117.  Pro/>.  To  obtain  a  general  formula  for  the  volume  of  a  solid 
bounded  by  a  surface  whose  equation  is  referred  to  polar  co-ordinates. 

Let  the  volume  be  divided 
into  elementary  wedges  such 
as  6^1  Z>,  CO  by  planes  drawn 
through  the  axis  OC.  Let 
each  wedge  be  subdivided 
into  elementary  pyramids, 
such  as  FGDEO,  by  coni- 
cal surfaces  generated  by 
the  revolution,  about  the 
axis  0(7,  of  lines  OD,  OE, 
&c.,  inclined  to  0(7  in  con- 
stant angles.  Finally,  let  each  pyramid  be  subdivided  into  elemen- 
tary parallelopipedons,  such  as  fd^  by  concentric  spherical  surfaces 
with  their  centres  at  the  origin  0. 

The  co-ordinates  of  a  point  d  are  Od  =z  r^  dODi  =  6,  and 
AOD-^  =  V ;  and  the  three  edges  of  the  elementary  parallelopipedon 
fd  are  dd^  =:  dr,  de  =  rdd,  and  dt  =  r  cos  &  ,dv;  the  last  expres- 
sion being  obtained  by  observing  that  when  the  line  OD  revolves 
around  the  axis   0(7,  the  point  d  describes  a  small  arc  dt  whoso 

23 


354 


IXTEGHAL    CALCULUS. 


centre  lies  upon  the  axis  OC,  and  whose  radius  is  the  perpen*- 
dicular  distance  of  d  from  that  axis,  and  therefore  expressed  by 
r.  smZOd  =  r  cos  &. 

Hence  the  volume  of  tho  parallelopipedon  will  be  expressed  by 

r2  cos  &dv  .  d&dr,     and     .' .  V  =  fffr'^  cos  &dv  .d(i  .dr  .  .  .  .  (1 ) 

will  be  the  required  formula  for  the  entire  volume. 

The  first  integration,  if  performed  with  respect  to  r,  while  v  and  & 
remain  constant,  will  give  the  sum  of  the  parallelopipedons  con- 
tained in  the  pyramid  DEFGO^  the  limits  of  the  integration  being 
r  r=  0  and  r  =  01)  =  F{vJ). 

A  second  integration  with  respect  to  &,  while  v  remains  constant, 
will  give  the  sum  of  the  pyramids  contained  in  the  wedge  G^D-^CO, 
and  the  third  integration  with  respect  to  v  will  give  the  sum  of  the 
wedges  which  constitute  the  entire  volume. 

118.   1.  The  hemisphere  with  radius  equal  to  a. 

Here  the  limits  of  the  integrations  are 

r  —  0  and  r  =  a,    &  =  0  and  &  =  ~^,  v  =  0  and  v  =  2if. 

• .  F  =  ///r2  cos  ^.dvMdr  =  i  ffr^  cos  &.dvd^  =  I  a^ff  cos  &.dv.d6 

o  o 

=  - a^/sin  &.dv=z-a'^.  sin  - 'rcfdo  =z  - a^v  =  - a^.  S-r  =  -  'Ko.^. 


3 


3 


2.  The  volume  cut  from  a  sphere  whose  radius  is  a,  by  a  cylindei 
with  a  circular  base  w^hose  radius  =  ^,  the 
centre  of  the  sphere  being  on  the  axis  of  the 
cylinder. 

Here  we  shall  have  for  half  the  required 
volume  01  ABODE, 

Iv  =/// 7-^  cos  &.dvd&dr, 

the  limits  of  mtegration  being 


£ 

a 

te 

c 

^ 

A 

^ 

0 

B 

J 

CUBATURE   OF  VOLUMES.  855 

Igt  r  =  0  and  r  =  01=  b  sec  &, 

6=0  and  4=cos-1-j     v  =  0  and  v  =  2if, 
a 

2d.    r=0  and  r=a,    ^=cos-i-  and  d  =  -<r,    «;  =  0  and  v  =  2*. 

The  first  set  of  limits  give 

Cffr^  cos  ddv.dddr  =  -ffr^  cos  6dv.d3  =  -  ffb^  sec^^  cos  &dv .  d^ 
o  o 

=  1 63//sec2^ .  6?i'c?^  =  i^Vtan  a .  dv 


=  -i^tan  I cos-i  -]fdv 


=  h-^  tan/tan-i^— jv  =  ^IP-s/a^-h^ 

A.nd  the  second  set  of  limits  give 

fffr'^(X)s6dv,d^dr=  -ffr^  cos d.dv.dd  =  -a^ffoos&dv . cW 
o  o 

= -a^/sin^  .c?v 
o 

=  -a^  sm  -  cr/rf?;  —  -  a^  sm  |  sm— ^  *^ ifdv 

-H'-^)-=i-('-^^> 

.•.lF=:|4a3-(a2-62)^/^2i:p]  and   F=^'^ra3-(a2- 62)^1. 


PAKT  III. 

INTEGRATION   OF   FUNCTIONS   OF  TWO   OR  MORE 
VARIABLES. 


CHAPTER  I. 

INTEGRATION    OP    EXPRESSIONS    CONTAINING    SEVERAL    INDEPENDENT 
VARIABLES. 

119.  When  a  differential  expression,  containing  two  or  more 
Independent  variables,  can  be  obtained  directly  by  differentiating 
some  function  of  those  variables,  it  is  said  to  be  an  exact  differentio.l. 

Thus  xdy  -f  ydx  is  an  exact  differential,  being  equal  to  d(xy) ;  so 
also  is  Zx'^dy  —  Zydx  -f-  Qxydx  —  ^xdy,  being  equal  to  d(^x'^y—Zxy) ; 
but  x'^dy  —  Zydx  is  not  an  exact  differential,  there  being  no  expression 
which,  when  differentiated,  will  produce  that  proposed. 

120.  If  a  differential  be  exact,  its  integral  can  be  determined  in 
all  cases  by  methods  which  will  be  explained,  but  we  shall  first 
establish  whereby  to  distinguish  exact  differentials. 

121.  Prop.  To  determine  the  conditions  which  indicate  that  any 
proposed  differential  is  exact. 

Let  the  proposed  expression  be  Pdx  +  Qdy^  in  which  P  and  Q 
may  be  functions  of  one  or  both  variables. 

If  this  expression  be  the  exact  differential  of  some  function  u  of 
X  and  y,  we  shall  have 

du  =  Pdx  -\-  Qdy (1). 


FUNCTIONS  OF  SEVERAL   VARIABLES.  367 

But  by  the  general  process  for  differentiating  a  function  of  two 
independent  variables,  we  have 

du=.^^dx-^^^dy (2). 

dx  dy   ^  ^  ^ 

And  since  (1)  and  (2)  mjist,  from  the  nature  of  the  supposition,  be 
identical,  the  following  conditions  will  exist,  viz.  ; 

^=^ (^)'  «4: (^)- 

Now  differentiate  (3)  with  respect  to  y,  and  (4)  with  respect  to  «, 
and  there  will  result 

dP^_  d'^u  dQ  _  dH 

dy  ~  dxdy  dx  ~  dydx 

But  it  has  been  shown  that  the  result  of  differentiating  «,  with 
respect  to  x  and  y,  successively,  is  the  same,  without  reference  to 
the  order  of  the  differentiations,  or  that 

dhi__dhi_  dl^_d^ 

dxdy  ~  dydx      '    '  dy        dx  '' 

Hence,  when  the  proposed  differential  Pdx  -f  Qdy  is  exact,  the 
condition  (5)  will  be  fulfilled.  The  converse  is  equally  true,  as  will 
appear  fully  when  we  attempt  to  integrate  such  expressions,  and 
hence  the  condition  (5)  is  called  the  test  of  inlegr ability. 

122.  Now  let  the  proposed  expression  be  Pdx  -j-  Qdy  -f-  Rdz^ 
involving  three  independent  variables. 

If  this  be  an  exact  differential  of  some  function  u  of  a;,  y,  and  ^,  then 

fifii  fi'ii  (i^i 

du  z=  -— dx  -\-  -r- dy  -^  -r- dz  =  Pdx  -\-  Qdy  4-  Rdz  ; 
dx  dy  dz 

whence 


du              du 
^-Tx!     ^-Ty' 

R  =  -—^  or  by  differentiation, 

dP       dH      dP       d-^u 

dQ        dH       dQ       d-'u 

dy  ~~  dxdy''     dz  ~~  dxdz^ 

dx  ~  dydx      dz  ~  dydz* 

dR       dhi 

dR        d'^u 

dx  ~  dzdx' 

dy       dzdy 

858  INTEGRAL  CALCULUS. 

cPu         d'^u        dH         d^u        d^u         d^u 
dxdy  ~  dydx'     dxdz  ~  dzdx*     dydz       dzdy 

Hence  we  have  three  following  conditions  of  integrability, 

dP_dQ     dP__dB^     ^_^ 

dy  ~  dx*      dz  ~  dx^      dz  ~  dy 

Similarly,  if  the  expression  were  Pdx  +  Qdy  +  J^dz  +  Sds  -j-  &c., 

involving  n  independent  variables,  there  would  be  -  n  (w  — 1),  con- 
ditions  of  the  forms 

dP__dQ      dP_dE     dP_dS_  dQ_dB_     dQ_dS 

dy  ^  dx^      dz  ~  dx       ds  ~   dx  ^       '    dz  ~   dy  ^      ds  ~  dy 

123.  1.  Is  a^ydx  +  x^dx  -\-  Pdy  +  a^xdy  an  exact  differential  1 
Here  F  =  a'^y  +  ^^  and   Q  —  b^  -^  d^x. 

dP         ,      dO         ,  dP        dO        ,    , 

.  * .  — r—  =  a^,    — r—  =  a^,    .  • .  — ;—  =  — -^  and  the  expression  if 
dy  dx  dy         dx 

integrable. 

dx  dy  xdy 

2.  Is Y  H T  ^^  exact  differential  ? 

Here         P  =.  (a:2  +  y-^y  and   Q  =  y-\\  -  (a-2  -f  y'^j^x'], 
^  =  -  y  (a;2  +  y')^ ,  ^  -  y-'[-{x^-^f)^+x\x^+y^)\ 

or  —7-=  —  y(^^  +  2/^)      =  — p,  and  the  expression  is  integrable 

ctx  ^y 

3.  Is  Zxdy  —  Ay'^dx  an  exact  differential  % 

and  since  8y  and  3  are  not  equal,  the  expression  is  not  integrable. 

124.  Prop.  To  oblain  a  general  formula  for  the  integration  of  the 
form  du  =  Pdx  +  Qdy^  when  the  condition  of  integrability  is 
satisfied. 


EXACT  DIFFERENTIALS.  369 

Since  the  term  Pdx  has  resulted  from  the  differentiation  of  the 
function  m,  with  respect  to  x  only,  y  being  regarded  as  invariable,  it 
follows  that  u  will  be  obtained  by  integrating  Pdx  with  reference  to 
X  alone ;  but  as  u  may  have  contained  terms  involving  y  alone, 
which  terms  necessarily  disappear  in  a  differentiation  with  reference 
to  x^  we  must  complete  the  integration  of  Pdx,  not  as  usual  by 
adding  a  constant  C,  but  by  adding  a  quantity  Z",  which  is  some 
unknown  function  of  y  and  "^constant,  and  we  thus  provide  for  the 
reappearance  of  such  terms  as  may  have  disappeared  in  the  first 
differentiation.     Thus  we  get 

u  =  fPdx+  F,  .  .  .  .  (1), 

in  which  the  value  of  Y  remains  to  be  determined. 
Differentiating  (1)  with  respect  to  y,  there  results 

du      d  f  Pdx      dY        ^        ^^  _  n 
dy~       dy  dy'  dy~      ' 

dY       ^      dfPdx      dY^         1^      dfPdx\^ 

'-'1^  =  ^ — ^'  Ty''-{^ — drr 

and  by  integration 

This  value  reduces  (1)  to  the  fjrm 

u=fPdx  +  f[Q-^^']dy (2), 

which  is  the  required  formula. 

125.  It    ife    necessary   to   prove,    however,   that    the   coefficient 

dfPdx 

Q of  dy,  does  not  contain  x,  since  otherwise,  the  second 

ay 

integration  would  be  attended  with  the  same  difficulty  as  the  first. 

Differentiating  that  coefficient  with  respect  to  or,  we  obtain 

dQ      (PfPdx  _dQ  _d^fPdx  _dQ      dP 
dx  dydx     ~~  dx  dxdy  dx        dy 


860  INTEGRAL    CALCULUS. 

and  this  is  equal  to  zero  by  the  condition  of  integrability,  which  it 
«upposed  to  be  satisfied.  Hence  the  coefficient  of  dy  in  (2)  cannot 
contain  x. 

126.  This  proof  also  establishes  the  truth  of  the  converse  of  the 

.  .  .         ,         1         ,  ,.  .       dP      cW    . 

iirst  proposition,  viz  :  that  when  the  condition   -r-  =  -r—    is    satis- 
fy       ao; 

fiod,  the  integration  is  possible. 

127.  By  a  similar  process  we  obtain  a  second  formula 

„  =  /e.,+/[p-^f].. (3). 

in  which  the  coefficient  of  dx  does  not  contain  y. 

Cor.  If  there  were  given     du  =  Fdx  +  Qdy  -\-  Rdz^     we  would 

write 

u  =  fPdx  -f  F, 

hi  which  F  is  a  function  of  y  and  z. 

Then  differentiating  with  respect  to  y,  we  obtain 

dV  _d_u_  d  fPdx  _  ^  _  dfPdx 
dy  ~  dy  dy      ~  d'y     ' 

•••  Ty''  =  L^--VJ^'' 

and  by  integrating  with  respect  to  y  and  adding  a  function  Z  of  «, 
we  get 

Now  differentiating  with  reference  to  z,  we  obtain 

dZ_du      dfPdx      d/Qdy      ±[  f^ff^.l 
dz  ~  dz  dz  dz  dz\J       dy  J 

in  which  the  coefficient  of  dz  is  independent  of  x  and  y. 


EXACT  DIFFERENTIALS.  361 

128.  In  piactice  it  will  be  found  usually  more  convenient,  and 
always  more  instructive,  to  apply  the  method,  rather  than  the  form 
ula  explained  above ;  especially  where  there  are  three  variables. 

EXAMPLES. 

129.  1.  Integrate     du,  =  {Zx^  +  'ilaxij)dx  +  {ax^  +  Zif)dy, 
Here  P  =:  3^2  _f_  ga^iy,  Q  =  ax^  +  Sy^. 

dP       ^  dO        ,    ,  

— •  =  2ax  z=  — — ,  and  the  expression  is  integrable. 

But  f£dx  =  f(Sx^  -I-  2axy)dx  =  x^  -\-  ax^y, 

dfPdx           ^          ,      ^       dfPdx 
.  •.    -^ =  ax^,     and     Q ^— —  =  ax'^+Zy'^—ax'^=^y\ 

These  values  reduce  (2)  to  the  form, 

u  =  x^  -{■  ax'^y  +  f^'^dy  =  x^ -[■  ax^y  +  y^  +  (7. 
2.  Integrate     du  —  {Sxy^—x^)dx  —  {l-{-6y^  —  Sxhj)dy, 

P  =  Zxy-^-x\     ^=_(l  +  62/2-32-23/),      ^  =  62^^  =  ^. 

3  1 

u  =  fPdx  z=  /(3a:y2  —  x'-)dx  —  -a:V  _  ^3  ^  j; 

Z  3 

•*  •   1|^=  ^  ~  ^"^'^  =  -(1+  6^2  _  3a:2y)-  3;r2y  ^  _  1  _,.  Oj,» 

-'-    -rdy=  -dy^Qy^y,     and      F  =  -  y  -  S;/ -f  ft 

3  1 

•.     tt  =  ^a:2y2-3«^  — y  —  2y3+  a 


862  INTEGRAL  CALCULUS. 

3.  du  =  (sill  y  -\-  y  cos  x)dx  -f  (sin  x  -{-  x  cos  y)dy, 

dP  dO 

— -  =:  COS  y  +  COS  a:  =  -^  • 
dy  ^  dx 

u  =z  f  (sin  y  -\-  y  cos  a;)c?a;  =  a;  sin  y  +  y  sin  a;  +  ^, 

dY      du  .  .  ^       ^ 

• .   -^-  =  -; a;  cos  y  —  sin  a;  =  0,     .  • .   Y  z=.  C. 

dy       dy  " 

and  w  =  a;  sin  ^^  -f  2/  sin  a:  +  C. 

4.  rf„^-^+^^+     ^^'^ 


a  —  ^       a  —  z       («  —  2)^ 

rfP_      1      _(f^      rfP  _         y        _dR      dQ  _         x        _dR 

dy~  a  —  z~   doc!      dz       («  —  ^Y  ~  dx''      dz        («  —  ^Y  ~~  ^V 

*J  a  —  z       a  —  z  dv       dii       a  —  z 


dy       dy       a  —  z 
dz  ~  dz       (a—  zY 


Then        ^=^-,-f2^„  =  0,      .-.     Z  =  C. 


i2L  +  <7. 


130.  In  practice  the  preceding  process  may  be  abridged  by  first 
integrating  Pdx,  then  integrating  the  terms  in  Qdy^  which  do  not 
contain  x,  and  finally  integrating  those  terms  in  Bdz  which  do  not 
contain  either  x  or  y,  and  adding  the  results.  That  the  complete 
integral  will  be  given  by  this  process,  appears  immediately,  from  the 
consideration  that  the  integration  of  Pdx  necessarily  gives  all  the 
terms  in  the  integral  sought  except  such  as  contain  y  and  z  without  x. 
Hence  in  integrating  Qdy  we  must  not  consider  any  term  which  con- 
tains X.  as  otherwise  we  would  introduce  into  the  integral  new  terms 
containing  x.  Similarly  the  integration  of  the  selected  terms  in 
Qdy  gives  all  the  remaining  terms  except  such  as  contain  z  only,  and 
therefore  in  integrating  Rdz  we  must  neglect  all  terms  involving 
both  X  and  y. 


HOMOGENEOUS   EXACT  DIFFEKENTIALS.  363 

E,.        du  =  ^'^^  +  y^y  +  '^'  +  g^^--4i  +  zdz  +  fdy. 

This  satisfies  the  conditions  of  integrability,  and  by  taking  the 
terms  in  Pdx  we  get 

Now  taking  the  terms  in  Qdy  which  do  not  contain  ar,  we  get 
fQdy  =  fyMy  =  -y\ 
and  finally  taking  the  terms  in  R  which  do  not  contain  x  nor  y, 
J  Rdz  =fzdz  =  -  2^. 

.  • .  I*  =  (;r2  +  2/2  +  ^2)4  4.  tan-i  ^  +  1 3^3  ^.  ^  ^2  +  (7. 


Homogeneous  Exact  Differentials. 

131.  Although  the  methods  of  integration  just  explained  apply  to 
all  exact  differentials,  yet  another  and  simpler  process  can  be  used 
when  the  expression  belongs  to  the  class  called  homogeneous.  A 
differential  expression  is  said  to  be  homogeneous  when  the  sum  of 
the  exponents  of  the  variables  is  the  same  in  the  coefficient  of  every 
term.     Thus 

ax^dx  --  by'^dy 


xdy  ■\-  ydx,     x^zdx  +  xz^f^x  —  xyzdy,     and 


(:.3_^y3)ti 


are  homogeneous  differentials.    The  degree  of  the  terms  is  estimated 
by  this  sum  of  the  exponents  ;  thus  in  the  first  expression  it  is  1,  in 

the  second  it  is  3,  and  m  the  third  it  is  2 —  =  —  --• 

10  lo 


864  INTEGRAL    CALCULUS. 

132.  Prop.  If  an  exact  differential  be  homogeneous,  and  the  terms 
of  any  degree  except  —  I,  its  integral  may  be  obtained  by  simply 
replacing  dx^  dy^  and  dz^  &;c.,  by  ar,  y,  s,  &c.,  respectively,  and  di- 
viding the  result  by  w  -f-  1,  when  n  denotes  the  degree  of  the  terms. 

Proof.  Let  da  =  Pdx  -f  Qdy  +  Pdz  -\-  &c.,  be  homogeneous  and 
exact,  in  which  P,  Q^  B,  6zc.,  are  algebraic  functions  of  x,  y,  z,  &c., 
of  the  degree  n. 

This  must  have  resulted  from  the  differentiation  of  a  homogeneous 
algebraic  function 

u  =  P,x+  Q,y  4-  /2^2  +  &c (1), 

of  the  degree  ?i  -f  1,  since  differentiation  diminishes  by  unity  one 
of  the  exponents  in  the  term  differentiated  at  every  step. 

Put  y  =  y^x,  z  =  z^x,  &;c.,  and  substitute  in  P^,  Q^,  i?„  &c., 
which  quantities  contain  x,  y,  z,  &;c.,  involved  to  the  n*^  degree. 
Replace  also  y  by  y^x,  z  by  z^x,  (fee,  in  (1)  ;  then  each  term  in  the 
value  of  u  will  contain  the  factor 

x''+\     and     .'.u  =  P2^«+i (2), 

in  which  Pg  is  a  function  of  y^,  z^,  &c.,  but  does  not  contain  x. 
Differentiating  (2)  with  respect  to  x  we  get 

~  =  (n+  l)P,x.  ....  (3). 

A  similar  substitution  in  the  value  of  du  gives 

du  =  Pdx  +  Qd{yix)  +  Ed{zyx)  +  &c (4) ; 

dii 
and  therefore  the  partial  differential  coefficient  — -  derived  from  (4) 

by  differentiating  the  products  y^x^  z^x,  &c.,  with  respect  to  x  only,  is 
~  =  P+«yx  +  i?^i,&c.....(5). 

Multiplying  (3)  and  (5)  by  x  and  equating  the  results,  we  get 
(n  -f  1)  P2:c«+i  =  Pa;  +  Qy^x  +  JRz^x  -  Px  +  Qy  +  Ms  ^  &c. 


HOMOGENEOUS  EXACT  DIFFERENTIALS.  365 

2  n  +  1 

fts  stated  in  the  enunciation. 

133.  When  n  =  —  I,  this  formula  would  make  u  z=z  co .  In  this 
case  it  is  easily  seen  that  the  formula  ought  not  to  be  applicable, 
because  it  is  not  then  true  that  the  desired  mtegral  is  an  algebraic 
function  of  the  degree  n  -\-  1  ]  but  on  the  contrary  it  is  tran- 
scendental. 

1.  To  integrate  du  =  {2y^x  -f  3y^)dx  +  {2x^y  +  9xy^  +  Sy^)dy, 

dP  dO 

—  =  4.yx+  9y2  =:.  _^. 
dy  -^  ^         dx 

.  • ,  the  differential  is  exact ;  it  is  also  homogeneous,  and  since 

»  =  3  or  w  +  l  =4,    w  =  ^^4^  +  <^=y^-^'+3^^«+2y*  +  C. 

n  -\-  \ 


dP     1     dq 

dP           y       dR 

dq      2y-x  __dR 

dy    ~  z  ~  dx' 

dz'^       z^~  dx' 

dz             z'^           dy 

. ' .  The  differential  is  exact ;  and  being  also  homogeneous  and  of  the 
order  0,  we  have  w  +  1  =  1. 

xy  —  2y2       y2z  —  xyz 
1 -9 r  (^ 


Px+qy-\-  Rz 

+  0  = 

yx 
z 

+ 

n-j-  1 

-p-i^'^a. 

134.  When  there  are  three  or  more  variables,  the  application  ot 
the  test  of  integrability  will  often  be  very  troublesome.  In  such 
cases,  if  the  differential  be  homogeneous,  it  will  be  found  more  con- 
venient to  apply  the  preceding  process  as  though  the  differential 
were  known  to  be  exact,  and  then  to  ascertain,  by  actual  trial, 
whether  the  given  expression  can  be  reproduced  by  differentiation. 


INTEGRAL  CALCULUS. 
{^^/x]f—z)dx        xdy        2{x^—Z'\/x)ds      y/xdz      . 

This  being  homogeneous  and  of  the  degree  —  -,  its  integral,  if 
possible,  must  be 

'2{'l)/xy—z)x         'Hxy         ^[x^/y^zy/x)s       "Hz-y/x 
^  _ __. ^ 1 f_  i,^ 

26-2  y^  '^^'^Vy  *  * 

~ 2.r«/  +  zor  —  xy   -f- Axy  —  Az3^  +  2zx    .p_^y  —^^    ,p 

This,  differentiated,  gives 

(y^-l^f^^        f^dy      2{xy^-zx^)ds      hz 

^^  -  -2  +  -2^  -  ,3  -         ,2    ' 

which  is  identical  with  the  proposed  expression  (1). 

It  must  be  distinctly  understood  that  in  the  differential  expressions 
here  considered,  the  variables  x,  y,  z,  (fee,  are  wholly  independent 
of  each  other.  If,  then,  the  conditions  of  integrability  be  not  ful- 
filled, the  integration  must  be  impossible,  since  there  is  no  relation 
between  the  variables,  by  the  aid  of  which  we  might  hope  to  trans- 
form the  given  differential  into  another  of  an  integrable  form. 

It  would  be  otherwise  if  a  relation  between  the  variables  were 
given  in  the  form  of  a  differential  equation,  such  Pdx  +  Qdy  =  0. 

Here  the  form  of  the  first  member  may  be  greatly  modified  by 
the  introduction  of  a  variable  factor  (or  by  other  methods),  and  thu^ 
(he  integration  may  be  facilitated. 


CHAPTER    II. 

DIFFERENTIAL    EQUATIONS.  • 

135.  A  differential  equation  between  two  variables  x  and  y  is  a 
relation  involving  one  or  more  of  the  differential  coefficients  such  as 

g.g,g,^,  .„.„,,..„  (If.  g)-,(g)U. 

Such  equations  are  arranged  in  classes  dependent  upon  the  07'der 

and  degree  of  the  differential  coefficient.     Thus,  when  the  equation 

„  dy    d'^y  d^y    .    .        . ,       . 

involves  only  the  first  powers  oi  — ,  -—  •  •  •  •  — ^,  it  is  said  to  be 

of  the  n^^  order  and  \st  degree. 

When  it  contains  only  the  powers  of  the  1st  differential  coefficient, 

viz.:   -r,  \~\  •  •  •  .  |-;^|  ,  it  is  of  the  \st  order  and  n*^  degree, 
dx    \dxf  \dxf 

And  when  it  contains  the  n^^  powers  of  one  or  more  differential  co- 
efficients, and  a  coefficient  of  the  m^^  order,  the  equation  is  of  the 
n*'^  degree  and  m*^  order. 

136.  The  resolution  of  a  differential  equation  consists  in  finding 
a  relation  between  x  and  y  and  constants.  This  relation,  called  the 
primitive,  must  be  such,  that  the  given  differential  equation  can  be 
deduced  from  it,  either  by  the  direct  process  of  differentiation,  or  by 
the  elimination  of  a  constant  between  the  primitive  and  the  direct 
differential  equation.  Hence  the  same  primitive  may  have  several 
differential  equations  of  the  same  order.     Thus  the  equation 

a^/  +  6.r  +  c  =  0  .  .  .  .  (1) 

gives  by  differentiation  a  -^  -f  6  =  0  .  .  .  .  (2), 


868  INTEGRAL  CALCULUS. 

and  by  elimination  between  (1)  and  (2)  we  get  the  indirect  differen- 
tial equations 

bx  ■—-  -\-  c- 6y  =  0  .  .  .  .  (3),  when  a  is  eliminated ; 

and         ay  •\-  c  —  aa;—  =  0  ,  .  .  .  (4),  when  b  is  eliminated. 
cix 

In  each  of  the  equations  (2),  (3),  and  (4),  the  variables  are  con 
hected  by  the  same  relation  as  in  (1),  which  latter  is  their  common 
primitive. 

131.  As  the  integration  of  differential  equations  can  be  effected 
in  comparatively  few  cases,  it  is  found  convenient  to  arrange  them 
in  the  order  of  the  difficulties  presented,  commencing  with  the  sim- 
plest oase. 

Differential  Equations  of  the  First  Order  and  Degree, 

1?3.  These  are  of  the  general  form  P+(?-^=:0  or  Pf/i:H-  Qd>j=0, 

in  which  P  and  Q  may  be  functions  of  both  x  and  y.  The  integra- 
tion will  obviously  be  possible,  by  the  method  applied  to  differential 
expre&sioiifi,  whenever  Fdx  +  Qdy  is  an  exact  differential,  and  the 
required  solution  will  be  of  the  form 

F{x,y)=:  C, 
where  C  is  an  arbitrary  constant. 

139.  Again,  the  Jntegration  can  be  effected  whenever  the  separa- 
tion of  the  variables  is  possible,  that  is,  when  the  equation  can  bo 

reduced  to  the  form 

Xdx  +  Ydy  =  0, 

where  X  is  a  function  of  x  only,  and  Y  a  function  of  y  only. 
The  form  of  the  solution  will  then  be 
Fx  -\-(pyz=  C, 
which  requires  only  the  integration  of  functions  of  a  single  variable. 


DIFFERENTIAL  EQUATIONS. 

The  separation  of  the  variables  is  possible  in  several  casea. 

140.  Cane  \st.  Let  the  fljrm  be 

Ydx  +  Xdy  =  0, 

in  which  the  coefficient  of  dx  contains  only  y,  and  that  of  dy  contains 
only  X. 

Divide  by  XY,  the  product  of  the  two  coefficients,  and  there  will 

result 

dx  ,   dy       .....    .  ...  , 

-~  -f-  ~  =z  0,  ni  which  the  variables  are  separated. 

141.  Ex.  Given    (1  -j-  y'^)dx  —  srdy  =  0,    to  find  the  primitive 
relation  between  x  and  y. 

Divide  by   (1  +  y")-^  >  then 

dx  dy  ^  ^  i-  t  ^ 

which  is  the  required  relation. 

142.  Case  2d.  Let  the  fjrm  be 

XYdx  4-  Xi  Y^dy  =  0, 

in  which  each  coefficient  is  the  product  of  a  function  of  x  by  anothei 
function  of  y. 
Divide  by  Xi  F, 

.  • .   -v^  -| ~~  =  0,  and  the  variables  are  separated. 

A^       r 

143.  Ex.  Determine  the  primitive  of 

(1  —  xfydx  —  (1  +  y>Vy  =  0. 
'    Divide  by  x^y, 

(I  —xf  ^         1  +  y  .         ^  dx      2dx      ,      dv       , 

...    L___I,;^___^^2,  =  0,  or,  --  —  +dx--j.^dy:=.0, 

.*. 2\ogx+x—logy—yz=C. 

24 


370  INTEGRAL   CALCULUS. 

144.  Case  3c?.  Let  the  proposed  equation  be  homogeneous,  or  d^ 
the  form 

Put  y  =  xz,         then         dy  =  xdz  +  zdx^ 

und  by  substitution 

^n+m^^OT  J^  aZ'^-'^  4-  62'«-2  ....    -{-  pz'^-'^)dx 

dx  {ez^  +fz'^-^  4-  gz"^-"^  .  .  .  .  +  5'2:'"-»)(/2 

=  0  and  the  variables  are  separated. 

145.  JEx.  To  find  the  primitive  of 

x'^dy  —  y'^dx  —  iryc?;c  =  0. 
Put  y  =  xz^         then         c/y  =  arc/^  4"  ^dx. 

.  • .   x^(xdz  4-  ^c?ic)  —  x^z^dx  —  x^zdx  =  0, 

dx      dz       ^  ^         ,  1        ^ 

.  • . 5  =  0,         and         log  X  4-  -  =  c7: 


or,  by  restoring  the  value  of   z  =  -^ 


X 
log  X  -\-  -  z=z  C. 

y 


2.  xdy  —  ydx  =  dxyx'^  —  y^. 

y  z=xZy         then         a;(a:(/2;  4-  ^(/a;)  —  xzdx  =  a:(l  —  g^)  V^r^ 

.  • .   —  = Y'         •   •  log  a:  =  sni-^2  4"  C, 

146.  The  same  method  of  transformation  may  be  extended  to 
such  differential  equations  as  involve  any  function  of-  unmixed  with 


DIFFERENTIAL   EQUATIONS.  371 

the  variables,  provided  the  equation  would   be   otherwise  homo 
geneous. 

_y 
-Er.  xydy  —  y^dx  =(a;  +  vY^  '^dx. 


Put  y  =  xz^  or,  -  =  ^j 


.  •,    x'^z(xdz  +  zdx)  —  x'^z^dx  =  a;2(l  -f  z^e-'dx^ 

dx         e'zdz              ,       ,                 «*       .    >^ 
.  • .   —  =  ..    ,     .,,      and       log  X  = h  C/. 

?!.  y 

.•.loga:=-^+C  =  -^  +  C7. 

147.  Case  \ih.  Let  the  form  be 

{a  -{•  hx  -\-  cy)dx  +  («i  +  h-^x  +  c-^y)dy  =  0. 

Put       a  -|-  6a:  4-  cy  =  v,         and         aj  +  b-^x  +  c^y  =  u. 

Then       c?«;  =  6c?^  +  cdy,         and         c?w  =  b-^dx  +  c^cfy, 

and  by  elimination 

_         c-,dv  —  cdu          ,     .    M    1         T         bdu—h.dv 
ax  =  —. — ' — - — ,     and  smiilarly      dy  =  — -^ — • 

.  • .    By  substitution     v(c^dv  —  cdu)  +  u{bdu  —  ^^c^v)  =  0, 
which  is  a  homogeneous  equation.  .    . 

148.  This  method  fails  when  bc-^  —  b-^c  =z  0,  because  the  attempt 

to    eliminate    either   dx    or    dy   causes    the    other    to    disappear 

be 
also.      But   since  we  then   have   Cj  =  ~,  the  proposed  equation 

reduces  to 


b  c 
{a-\-bx-^  cy)dx  +  (aj  +  b^x  +  -^y)dy  =  0  . (1). 

Put     bx  -\-  cy  =  z^  then  x  =  — 7—^,  and  dx  = j — -\ 

0  0 


872  INTEGRAL   CALCULUS. 

and,  by  substitution  in  (1), 

,      ,     .dz  —  cdy       t       ,    h.z\  . 

(«  +  ^)  — y-^  +  («i  +  -l-yy  =  0. 

ca  -\-  cz  —  a^b  —  byZ 
an  equation  in  which  the  variables  are  separated. 
149.  Ex.  Find  the  primitive  of 

{\^x  +  y)dx  +  (1  +  2^  +  ^)dy  =  0. 
Put  I  -j~  X  -\-  y  =  v^  1  -\-2z-^  Sy  =  u'j  then 

dx  -\-  dy  z=:  dv,    and   2dx  -j-  oc?y  =  c^w, 

,  • .  c?a;  =  3c?v  —  du,  and  Jy  =  t/i*  —  2dv. 

.  • .  v{Bdv  —  du)  +  2^(g?i/  —  2dv)  =  0. 

Now  put  u  =  r?^,  then  du  =.  rdv-\-vdr^  and,  consequently, 

v{Zdv  —  rc?v  —  vt?/)  +  rv[rdv  -\-  vdr  —  2dv)  =  0, 

••■'--^^[('-l)ve+>-[4f-|)]=« 


or 


llog  [«^  -  Suv  f  3,.=]  +  -L  ta„->  fA  (H^ifAnl  ^  c 


150.   Case  5^A.  Let  the  form  be  dy  +  Xydx  =  -X\(/ar (l)j  ^n 

which  X  and  Xj  are  functions  of  x. 

The  peculiarity  of  this  form  is  that  no  power  of  y  except  the  first 


SEPARATION    OF    THE   VARIABLES.  373 

enters   into  it,  and    for   this   reason   it   is   usually  called  a  linear 
equation.     Its  solution  is  always  possible. 

Put  y  =  X^  where  X2  is  an  arbitrary  function  of  x,  which  may 
be  so  assumed  as  to  facilitate  the  integration ;  and  z  sl  new  and 
undetermined  variable.     Then 

t/y  =  Xjcfe  -f  zdX2,  and   (1)  can  be  reduced  to  the  form 

Xr^dz  -f  '^dX^  +  XX^dx  z=  X^dx (2). 

Now  let  Xg  be  determined  by  the  condition 

zdX^  =z  X,dx (3), 

and  (2)  will  become      Xg^/^  +  XX.^dx  —  0. 

.  • .  =  —  Xdx     and     log  2  ==  —  fXdx,  .' .  z  =z  t        '. 

This  value,  substituted  in  (3),  gives 

e-'SXdzdX2  =  X^dx     or     dX^^  ef^^^X^dx. 
.• .  Xj  =  JeS^d^x^dx,  and  y  =  0X2  =  e-fxdxfefxdzx^dx  ...  (4), 
which  is  the  required  relation  between  x  and  y. 

151.  Let  there  be  now  taken  the  more  general  form 

dy  4-  Xydx  =  X^y^dx (5). 

This  is  easily  reduced  to  the  linear  form  (1).     For  put 
m  —  n  -{-  \     and     z  =  y-«. 

i/^dz 

Then  dz  =:  ^  ny-^-Hy  or  dy  =z  — 

n 

Substituting  this  value  of  dy  in  the  equation  (5),  and  reducicgj 
we  get 

dz      Xdx 
1 ~  =  Xydx^  or  dz  —  nXzdx  =  —  nX^dx  .  . ,  .  (6), 

which  becomes  identical  with  (1)  when  we  replace 

2  by  y,  —nX  by  X    and     —  w  JC^  by  Xj. 


874  INTEGRAL   CALCULUS. 

...  z  =  -\-=-  ne^fxdxfx^e-r^fxdxdx (7) 

Cor.  In  forming  the  integral  fXdx^  it  will  be  unnecessary  to  add  a 
constant  C.  For  if  we  replace  fXdx  by  X^  +  (7,  the  formula  (4) 
will  become 

y  =  e-^3  -CfeXz  +  cx^dx  =  ^x.^-cfeX,  gC  x^dx  =  e-^^  fe^'  X^dx, 
in  which  the  constant  C  has  disappeared. 

152.  1.  To  determine  the  primitive  of  (1  -f  x'^)dy  —  yxdx  =  adx. 

yxdx 

1  +a;2       \-\-x 


Here  dy ^    ^   ^^  =  - — j — -  dx,  which,  compared  with  the  linear 


form,  gives 


X  =  —  — — — r    and    X^  = 


l+x^  '       l-i-x 

r.fXdx  =  -f- 


xdx         ,  1 

=  l0£f 


efxdx  —  e  log  [(i+x^rt  _  (1  _^  a;2)"^and  e-f^^""  =  (1  +  «^)^. 

,  • .  J,  =  (1  +  a:^)*  F— ^^  +  cl  =  ax  +  (7  (1  +  x^)*. 

2.  c?y  +  yt^a;  =  xy^dx. 

Here      X  =  1,  Xj  =  a;,   y"*  =  y^,   or   m  =  3,   and   n  =  2, 
and,  by  substitution  in  (7), 

or  I=:[(7e2x  +  aj  +  |]y2. 

3.  c?y  —  r ydx  =  6cfo*, 


SEPAKATION   OF  THE  VARIABLES.  376 

^'''  ^ = i^'  ^1  =  *•  •  •  •  /^<^^ = /r^  =  'OS  (1  -  "Y' 

e/Xdx  —  glog(l-a:)a_  ^1  __  a;)o^ 

fX^ef^^^ .  dx  =z  fh{\  -  xY  dx=z ^—-  (1  -  a;)«+i  +  C, 

tt  "P  1 

4.  Find  an  expression  for  the  sum  of  the  series 

X        x"^     ^       x^  x^ 

^  =  T  +  n  +  17375  + 1737577  +  ^''- 

Differentiating,  we  obtain 

dv  x"^  X^  X^  r3  7*5 

i=^ +7+0  +  075 +^^-=i+-^(^+o+il:5+^«-) 

=  1  -f  ^y.     .'.  di/  —  xydx  =  dx^  a  linear  equation. 
Also  fXdx  =  —  fxdx  =1  -\x\ 

,' .  y  =  e      (fe        dx),  the  desired  expression. 

153.  Case  6th.  Let  the  form  be 

dy  -f  Sy^i/a;  =r  ax^dx (1). 

This  form,  which  is  called  RiccaWs  equation,  involves  only  the 
second  power  of  y.  Its  integration  has  been  effected  for  certain 
values  of  the  exponent  w,  but  a  solution  applicable  to  all  values  of 
m  has  not  been  discovered. 

154.  It  is  obvious  that  when  m  =  0  the  equation  (1)  will  be  inte- 

grable,  for  then 

dy 

dy  +  by'^dx  =  adx.      .  * . ^-r-r  =  dx, 

a  —  by^ 

and  the  variables  are  separated. 


376  INTEGRAL   CALCULUS. 


ihus  we  have    2a   U   ax  =z r 


.  • .  2a  6   ^  =  log  — 4-  C. 

6  y  —  a 

Next  suppose  m  to  have  some  value  other  than  0. 

\         z 

Assume  y  =■  -, 1 — ^,  where  z  is  an  undetermined  variable,  but 

hx       x^ 

obviously  a  function  of  x.     Then  * 

dx       2zdx       dz  a/  ^       .     ^^     .   ^'\ 

Substituting  these  values  in  (1),  it  becomes 

dz       hz^dx  ,  ,        bz^  ,  ,„  ,  ,^^ 

-^  -\ =:  a^iFdx     or     dz  -\ dx  =z  ax^-^^dx  ....  (2). 

x^  a;*  x'^ 

Now  this  equation  (2)  will  be  homogeneous,  and  therefore  inte- 
grable  when  wz  =  —  2,  and  thus  a  new  integrable  case  is  found. 
Again,  if  w  =  —  4,  the  variables  x  and  z  can  be  separated,  for  then 

z^  dz  dx 

dz  -{-  b—dx  =  axr^dx.     .  • . — -  ==  —r. 

x^  a  —  bz^      x^ 

which  is  a  third  integrable  case. 

155.  To  obtain  others,  put 

-  =  y.     and     a:'"^^  _  -j. 
z  ' 

Then  dz=d—  =  -%     x-^^Ux=-^. 

Vi       y^  y«+3 

,  ax  \  m+Sj 

and  —  = — r  arj  dx^, 

x^       m  -\-  6    ^ 

Hence  by  substitution  in  (2), 


SEPARATION   OF  THE   VARIABLES.  377 

b                         a           J           ^         m  +  4 
Now  put =  a,,     — -  =  6,,    and — -  =  m, 

and  (3)  will  become,  after  reduction, 

^^1  +  b^yi^dx^  =  ai-Tj'"!  c/a;j (4), 

which  is  identical  in  form  with  (1).  Hence  (4)  must  be  integrable 
when  the  exponent  m^  has  either  i)f  the  three  values  0,  —  2,  or  —  4. 
Moreover  when  a  relation  has  been  obtained  by  integration  between 
z\  and  y^.  a  simple  substitution  will  give  the  desired  relation  be- 
tween X  and  y. 

We  have  therefore  to  examine  whether  by  assigning  either  of  the 
values  0,   --  2,  or   —  4  to  m-^,  any  new  values  of  m  will  arise. 

T^  wi  +  4  .  „  ^        ,  3m,  -f  4 

i3ut  m,  = -•    .'.  mm-,-{-3m-,=z  —m—%  and  m= ^— r- 

w  +  3  wij-f-  1 

.  * .  when         nil  =z  0,  m  =  —  4,  a  case  before  considered ; 
and  when       m^  =  —  2,  m  =  —  2,     "  "         " 

o 

but  when       m^  =  —  4,  m  =  —  -,  a  new  case. 

o 

Q 

Hence  Riccati's  equation  is  integrable  when  m  =  —  -  also. 

o 

.  156.  In  a  manner  entirely  similar  to  that  by  which  (1)  was  trans- 
formed into  (4),  may  we  transform  (4)  into  a  new  equation 

^1/2  +  hy2dH  —  o^x^dx^  ....  (5), 

,  .  ,  m^  +  4  -,  ,,       ^  Smg  4-  4 

m  which  m,  = ,     and  therefore     w,  = . 

^  mj  4-  3'  *  m^  4-  1 

And  by  repeating  the  process,  a  series  of  such  equations  may  be 
formed ;  so  that  it  will  be  possible  to  find  a  relation  between  x  and 
y  when  any  one  of  the  following  quantities  or  exponents  shall 
be  =  —  4  ;  viz. : 

m  -f-  4                        m,  +  4                        w„  +  4   „ 
m,  or  mi  = -,  or  mg  = ^— -,  or  m^  = ^— -,  &c. 


378                                     INTEGRAL    CALCULUS. 

But                    m  =  -  1 

when 

mj  =  —  4 

8 

n 

Wl2=  -4 

8 

(( 

7713  =   —  4 

Hence  by  successive  substitutions, 

12 

when 

wzj  ~  —  4 

16 

(( 

TWg    —     —   4 

&c. 


>  &c. 


Thus  Riccati's  equation  is  integrable  for  all  values  of  m  included  in 
the  series. 

_4.       _?      _I?      _i?      _?!?      _?1      A. 
3'  5'  7'  9'  11' 


157.  The  general   formula   for   these   numbers   is 


4n 


,m 


2n-  1 

which  n  is  any  positive  integer. 

To  prove  this,  suppose  one  of  the  numbers,  as  mg,  to  be  of  the 

form  required  —  ; ;   then  will  the  adjacent  terms  m^  and  mg 

be  of  the  same  form,  with  the  number  n  increased  or  diminished  by 
unity.     For  we  shall  have, 

4-3-^" 

3mo  4-  4  27. 

m-  =  — 


and 


2n  -  1 

both  of  which  forms  are  similar  to  that  given  above  as  the  general 
form. 


3mg  +  4 

^      "2/.  -  1 

4/1  +  4 
2n  +  1  ~ 

4(«  +  l) 

mQ  +  1  ~ 

1          ^'^ 

2(«  +  l)-l 

*      2w  — 1 

4          4. 
2/t-  1 

^8  +  4 

4n--4 

4(n-l) 

^8  +  3 

.          4.      - 

2/i  — 3~ 

2(/.-l)-l 

SEPARATION    OF  THE   VARIABLES.  879 

But  we   have   seen  that  one  integrable  case   is  that   in  which 

8  4.2 

m  =  —  -  =  —  ——' which  being  of   the  required  form,  the 

12 
adjacent  numbers  —  4  and ~  are  also  of  that  form ;  and  thence 

the  same  reasoning  can  be  extended  to  the  other  numbers  in  the  series. 
158.  Second  Transformation  of  BiccaiVs  Equation. — In  the  given 

equation 

dy  +  hy'^dx  =  ax^dx (1). 

put  y  =  — ,     and    a;"'+i  =  a?, 

m 

dy.             ^            dx.  ^       ^        X.   ^'^^dx-y 

then      dy  z= -j^,      x'^dx  = ~ ,      and     dx  =  - — r-rr' 

Also,    y^  =  — ,  and  therefore  by  substitution  in  (1), 

m 

y^'^{m  +  \)y^^       m  +  1      '' 

JNow  make      -•  =  6,, =  a,, ; — -  =  w,, 

and  the  equation  (8)  will  reduce  to 

dy^  +  ^i2/i'^^-^i  =  a^x^^idx-^ (9). 

which  is  of  the  same  form  with  (1). 

The  equation  (8)  will  evidently  be  integrable  whenever  m-^  has 
any  one  of  the  values  included   in  the  series  before  found,  that  is, 

4n 


^1 

or 

its  equal 

m  +  1 

- 

m 

4w 

-,  then  27nn  —  m 

2w  —  1 

m 

-f  1~ 

and 

4n 

,  • .     m  —  —  - — -— -.  . 

2n-  I 


But  if 


2n+.i 


880  INTEGRAL  CALCULUS. 

Hence  we  have  a  new  series  of  integrable  cases  corresponding  tc 

all  values  of  m,  included  in  the  formula  —  - — — -•   Thus Riccati's 

2n  -\-  I 


in 


equation  is  integrable  whenever  the  exponent  m  can  be  expressed  i 

4/i         ,  .  -    .  ... 

the  form  — r,  the  quantity  n  being  a  positive  integer. 

It  appears  also,  that  whenever  the  given  value  of  m  is  found  hi 

4/1 

the  second  series,  the  terms  of  which  have  the  form — ; — -,  an 

2/t  -j-  1 

application  of  the  second  transformation  will  transfer  m  to  the  first 

series,  and  then  the  repeated  application  of  the  first  transformation 

will  eventually  reduce  m  to  the  value  —4. 

159.  1.  To  integrate  the  equation 


dy  +  ifdx  - 

r  a^x   ^dx.  . 

...(1). 

Here 

m  =  - 

4 
"3"  " 

4.  1 
2.1  +  1" 

4n 
2«  +  1' 

and  m  is 

found 

in  the 

second 

series. 

.  • .     Put 

y 

1 

~yi 

and 

.-♦■^'  = 

.-*=... 

.  • ,    X  =  xf^,     dx  =  —  oxf^dx^, 
X     dx  =  —  Sdx,,     and     dy  =: J» 

y^ 

Hence  by  substitution  in  (1), 

_  ^1  _  -i  xr^dx.  =  -  Sa^dx., 

yi"     yi' 

or,  dy^  —  Za?y^Hx^  =  -  Zxf^dxy^ (2). 

Now  put  —  3a2  z=z  ij,     and     —  3  =  aj, 

then  dy-^  -{■  h^y^dx^  =  ayC^-^dx^ (3). 

4.1  An 


Her©  w,  =  —  4  = 


2.1-1  2«  —  1' 


SEPARATION    OF    THE  VARIABLES.  381 

•   •  ^^1  -  -  b,x,^      ^^  +  x,^'     ^'^'   -  b,x^  ^  x,^^l^ 
Hence  by  substitution 

cfej       b^z^dx^      a-^dx-^  dz-^       _     dx^ 

or,  by  replacing  a-^  and  Sj  by  their  values 

dz^  dx^  Sadxj        2^2^ 


3.  cfy  4-  y'^^^  =  ^x^dx.  .  .  (1). 

Here     m=— -=— .r r-     .  • .  Put  y=  7- + -|  =  "+ -f* 

3  27t  —  1  bx      x^      X     x^ 

Then  (/^j  +  z^xrHx  =  4x  ^dx. 

Now  make  gj  =  — ,     and     x  =x    =  x^.     Then 

-  ^^  +  -2  •  ^'  =  l^c^^i'     «^    ^2/1  +  ^^Vi^dx  =  3ari-*(/ar,.  .  (2). 
Vi        Vi      ^1 


Repeating  the  process  of  substitution  as  in  the  last  example,  we 
get 

ds^  4-  \'ilz^x{-'^dx^=:  Zxf^dx^    or, 


3c?ari  dz2 


i"  W  - 1 


INTEGRAL  CALCULUS. 


the  integral  of  which  is 


-  =  logc[^^J      or,    .-    [2^]=^^' 


c  =  e^ 


or. 


2x%z^  -  a:)-i  -  ^x^-  1 
S(;r  +  e/)  -  1  -  Qx'^  -  12a:~*-^ 


160.  If  the  proposed  form  be  thj  +  hi/x^dx  ~  c^«</a;,  which  differs 
from  the  form  just  considered,  in  having  a  power  of  x  in  the  second 
term  of  the  first  member,  it  may  be  readily  reduced  to  the  simpler 
form  by  making  x^dx  =  dz.     For  then 


a;r+i  =  (r  4.  1)2;,  and  a;«+i  =  (r  +  1)' 


r+l   ^H-1 


.  •  .  a;«c?a;  =:  [(r  +  l)zJ^Uz,  and  c/y  +  hyHz 

s—r 

=  c[(r  4-  1  )z]'^^dz  z=.  az'^dz, 
the  form  in  which  Riccati  equation  has  been  integrated. 


Of  the  Factors  necessa/ry  to  render  Differential  Equations 

exact. 

161.  The  cases  examined  above  embrace  the  principal  forms  in 
which  the  integration  is  possible  by  a  separation  of  the  variables. 
We  now  proceed  to  consider  those  in  which  the  first  member  of  au 


FACTORS  OF  DIFFERENTIAL  EQSjATIONS./ /  .        883-* 

equatn^n  Pdx  -^  Qdy  =  0  can  be  rendered  an  «:^act*<}ifirg*enttat^bv        ^^  ^ 
the  introduction  of  a  suitable  factor.  \      /^y  •>  .^       "^ 

162.  If  the  eqajition  Pdx  -\-  Qdy  =z  0  has  been  obtaineid  byiJl^AJt         (j 
aiflferentiation,  it  will  satisfy  the  test  of  integrability,  viz. :  ^  ^ 

dP  _  dQ 

dy   ~    dx   "* 

but  if  it  has  resulted  from  the  elimination  of  a  constant  between  the 
direct  differential  and  its  primitive,  that  condition  will  not  be 
satisfied,  although  the  same  relation  between  x  and  y  will  be  implied 
by  the  two  differential  equations. 

163.  Prop.  To  show  that   an   indirect  differential   equation  can 
always  be  rendered  exact  by  the  introduction  of  a  suitable  factor. 

Let  Pdx  -\-  Qdy  =0 (1) 

be  the  given  equation  which  has  resulted  from  the  elimination  of  a 

constant  c  between  the  primitive  and  its  direct  differential ;  and  let 

the  primitive  be  solved  with  respect  to  c,  giving  a  result  of  the 

form 

c  =  F{x,y) (2). 

Differentiating  (2),  c  will  disappear,  and  we  shall  obtain  an  equation 

of  the  form, 

F,dx  -^Q,dy  =  0 (3). 

Now,  since  (1)  and  (3)  contain  the  same  constants,  combined  with 
X  and  y,  and  since  the  same  relation  connects  x  and  y  in  the  two 

dv 

equations,  the  differential  coefficient  -j-  must  be  the  same,  whether 

derived  from  (1)  or  (3). 

•    ±--L-_£l.     or    £-A     and      •    ^-^ 

Hence,  if  we  multiply  (1),  the  first  member  of  which  is  not  an 

P       0 

exact  differential  by  -^  =  — i,  we  shall  convert  it  into  (3),  which  is 

-t  Q 

exact. 


384  INTEGRAL    CALCULUS. 

Cor,  If  it  were  possible  to  determine  this  factor,  the  integration 
of  every  differential  ec[nation  of  the  first  order  and  degree,  could  be 
effected  without  serious  difficulty,  but,  unfortunately,  the  difficulty 
of  discovering  this  factor  is  usually  insuperable. 

164.  Prop.  To  exhibit  the  condition  or  equation,  the  solution  of 
which  would  give  tiie  factor  necessary  to  render  any  proposed 
differentia!  equation  exact. 

Let  Pdx  -f-  Qdy  =^0  be  the  given  equation,  and  z  the  required 
factor. 

Then  Pzdx  4-  Qzdy  =  0,  and  the  first  member  will  be  exact. 

dPz        dQz       ,  .     ,  ,.  . 

.  * .  — 7 —  =  —7 — ,  the  requH'ed  condition. 

No  general  method  of  resolving  this  equation  is  known.  There 
are,  however,  several  particular  cases  in  which  the  factor  can  be 
found. 

165.  Prop.  To  show  that  when  the  factor  which  renders  an 
equation  integrable  has  been  found,  an  indefinite  number  of  such 
factoi's  can   be  discovered. 

Let  z  be  the  factor  first  found.     Put 

Pzdx  +  Qzdy  =  du. 

Multiply  by  Fu,  an  arbitrary  function  of  u,  and  there  will  result 

Fu .  Pzdx  4-  Fn  .  Qzdy  =  Fu  .du; 

and,  since  the  second  member  is  exact,  (containing  u  only)  the  first 
member  must  be  exact  also. 

.-.  z.Fu  =  z.Ff(Pzdx  -\-  Qzdy)  is  a  suitable  factor. 

166.  Prop.  To  explain  the  process  for  finding  the  required  factor, 
whei!  ihe  equation  can  be  separated  into  two  parts,  for  each  of  which 

a  sc]  .  r.'i,   ;■)  '..I-  can  be  found.     Let 

>  :  0 (•)  he  divisible  into  the  two  parts. 


FACTORS   OF  DIFFERENTIAL  EQUATIONS.  385 

{P,dx  -^  Q^dy)  +  {P^dx  4-  Q^dy)  =  0 (2), 

and  let  z^  and  Zy,  be  the  factors,  which  will  render 

P^dx  +  Qidi/  and  Pg^j;  +  Qzdy  separately  integrable. 
Put  Zy{Pidx  +  Q\dy)  =  du-^^    and    z^P^dx  +  ^2^y)  =  ^u.^ 
Then  s^i/^i^i  and  z^F^u^  will  also  be  suitable  factors  to  render  the 
two  parts  separately  integrable.     If  therefore  we  can  so  select  F^u^ 
and  i^2"2^^  ^^  ^^^^^  ^^®  condition 

either  of  those  factors  will  render  the  entire  equation  integrable. 
167.  1.  To  find  the  primitive  of 

adx      hdy       cx^dx  _  .  . 

^        y         y*    ~~    

This  can  be  resolved  into  the  two  parts 

adx   .   bdy  .  cx^dx 

h-^     and r-; 

X  y  y*    ' 

the  first  of  which  is  an  exact  differential,  and  therefore  0^  =  1  ;  and 
the  second  can  be  rendered  exact  by  the  factor  y*  =  stj. 

=  a  log  X  -\-  h  log  y  =  log  {x'^.y^). 

^2  =y  L~^2  (-  ^^)j  =/(-  ^w^)  =  ^^— p- 


Hence  we  must  endeavor  to  satisfy  the  condition  z-^F^u^  =  z^F^u.],^  or 
1  X.iP,  [log  (^«y»)]  =  y».  i^,(-  '^^. 

L^ —  )  =  .c*iii 

71  +  1/ 

in  which  k  and  Ar^  are  undetermined  constants.     Then 
25 


886  INTEGRAL    CALCULUS. 

a  condition  which  will  be  satisfied  by  making 

kb  =  b     and     ka  =  k,n.     or     k  =  1     and     Ar,  =  -• 

Hence  a?«</*  is  the  required  factor. 

Now  multiply  (1)  by  x°i/^,  and  there  will  result 

Q^o-i  yb(i^  _|_  ix'^yh-i  fly  _  cx^-\-n  dx  =  0, 

which  is  exact,  since 

dy  ^         dx        * 

and  the  required  solution  is 

a  -f-  ?t  +  1 
2,        •  -  a:fl?y  —  ydx  —  -  adx  =  0. 

This  can  be  resolved  into  -  xdy  —  ydx,  and  —  -  adx,  of  which  the 

first  will  be  rendered  exact  by  the  factor  z^  =  — ,  and  the  second 

xy 

18  already  exact,  givinjr  ^g  =  !• 

•  •  •  ^-  -Ily  G  ^'y  -  y'^)  =flj  -fi  =  ^^g  T 

And  we  must  satisfy  the  conditions 

Assume      ^,(log^^)=|-     .-.l/'.llog^^j^l. 


FACTORS  OF  DIFFERENTIAL  EQUATIONS.  887 


Hence  ~y  is  the  proper  factor. 


1   xdy      ydx       1    adx 

which  is  exact,  and  the  solution  is 

J^  +  i~+(?  =  0    or    y+&^  +  |  =  0. 

168.  Prop.  To  determine  the  factor  necessary  to  render  a  differ- 
ential equation  exact,  when  that  factor  is  a  function  of  one  variable 
only. 

Let  Pdx  -h  Qdy  =  0  ....  (1)  be  the  given  equation,  and  z  z=  Fx 
the  required  factor. 

.  • .  zPdx  -f  zQdy  =  0  will  be  exact,  and  therefore 

d(zP)       d(zQ)  .  ^  .  :,    ^      ^       dz      ^ 

-^, — -  =  -^ — - ;  or  since  z  does  not  contain  y,  and  therefore  -;-  =  0, 

dy  dx    ^  ^'  dy        ' 

dy  dx  dx  z        (^x^dy        dx  \ 

Here,  by  hypothesis,  the  first  member  does  not  contain  y,  and 

therefore  the  second  member  must  be  independent  of  y  also.     Con. 

eequently 

rVdP       d<J\dx  ,  A,     ,  .     ,      , 

log  z  =  /  I  — — -  I  —  =  9^;  and  z  =  e^i  the  required  value. 

169.  1.  Given         ydx  —  xdy  =  0 (1). 

Suppose  z  to  contain  x  only. 

dP_dy^_^       dQ  _d(—x)  _ 
dy       dy         ^      dx  ~~      dx 

[dP      dQldx  _       2dx 
dy        dx  jQ  ""         X 

...log«=-/?^  =  logi    and     ^  =  1. 


888  INTEGRAL  CALCULUS. 

Multiplying  (1)  by  —  we  get 

l^?^  =  0    and     1=0    or    y  =  C^. 

2.  The  linear  equation  di/  -\-  Xydx  —  X^dx  =  0. 

Suppose  z  —  Fx.     .•.-—  =  -^-^, ^-^-  ==  X    and   -7^  =  0. 

,  dy  dy  dx 

.  • .  log  2;  =  jXdx    and    z  =  c/^^^,    the  factor  sought. 

Multiplying  (1)  by  this  factor,  we  have 

efxdx^f^y  _|_  eS^<^^  Xydx  —  eS^^^  X^dx  =  0,  which  is  exact. 

Remark.  The  value  of  z  found  by  the  method  just  explained,  was 
obtained  by  assuming  that  a  factor  containing  x  only  can  be  dis- 
covered ;  but  since  such  factor  may  nut  exist,  it  will  be  proper  to 
apply  the  test  of  integrability  to  the  transformed  equation." 

170.  Prop.  To  determine  the  factor  necessary  to  render  a  homo- 
geneous differential  equation  exact. 

Let  Pdx  -f  Qdy  =  0 (1) 

be  a  homogeneous  differential  equation,  the  coefficients  P  and  Q 

being  each  of  the  7i"»  degree;  and  let  the  factor  z  be  of  the  m'* 

degree.     Then 
/  zPdx-\-zQdy  =  0 

will  be  exact,  homogeneous,  and  of  the  (m  +  w)'*  degree. 

Hence,  by  the  rule  for  integrating  homogeneous  exact  differentiial 
•xpressions,  we  have 

or,  since  C  is  arbitrary,  we  may  put  (w  +  h  +  1)  C  =  1. 


GEOMETRICAL  APPLICATIONS. 
1 


.  z  = 


is  a  suitable  factor. 


Px^  Qy 
Ex.  {xy  +  y^)dx  -  {x''  -  xy)dy  =  0 (1). 

Here  xy  -\-  y^  =  F    and     —  x^  -^  xy  =  Q. 

1  1 


.  •.  z 


Multiply  (1)  by  z. 


Px  -f  Qy        2iry2 

dx         dx        xdy   ,     %  _  /v 
2y  "*""2J~  2y2"^  27""    * 


389 


\    X  \  1  X 


Geometrical  Applications  of  Differential  Equations  of  the 
first  order  and  degree. 

171.  1.  Determine  the  curve  whose  tangent  PT  is  a  mean  pro- 
portional between  the  parts  ^7"  and  BT  ^,^_^^^ 

of  its  axis,  intercepted  between  the  tan- 
gent and  two  fixed  points  A  and  B. 
Place  the  origin  at  B^  and  put 

BD  =  x^    DPzzzy^     BA  =  a. 
The  equation  of  the  tangent  is  y  —  y^  =  —  —  •  {x  ~  arj), 


in  which  when 
•nd 


dx 


Vi 


dx^ 


=  BD-{-DT=lBT. 


dx^ 


And  >T^z=zPD^+DT^  =  y.^^+y  2^ 

dy^ 


390  INTEGRAL   CALCULUS. 

Hence,  by  the  conditions  of  the  problem. 

PT^  =  BTxAT,    or    y,2  +  y,2^ 

Reducing,  and  omitting  accents,  we  get 

y'^dy  =  x^dy  —  ^xydx  —  axdy  -\-  aydxj 

which  is  the  differential  equation  of  the  required  curve. 
This  may  be  written 

2xydx  —  xMy         ydx  —  xdy 

and,  since  both  members  are  exact,  we  get  by  integration 

y  -\ =  a  -  -}-  (7,     or     x^  ■\-  y"^  —  ax  —  Cy=() ;  or,  finally, 

which  is  the  equation  of  a  circle  whose  radius  is  -  ya^+  C^  j  and 

the  co-ordinates  of  whose  centre  are  -  a  and  -  C,  the  latter  co-ordi- 
nate  being  arbitrary. 

2.  Find  the  curve  in  which  the  subtangent  is  constant. 

Let  ari^i  be  the  co-ordinates  of  the  point  of  contact. 

_,  ,  dx-,  dy      dx 

Then,  subtangent  =  —  y-^  —  =  —  a,     or     —  =  — • 

X 

.-.  logcy  =  -^,       cy  =  e«. 
ft 

This  is  the  equation  of  the  logarithmic  curve. 

8.  To  find  the  curve  in  which  the  subtangent  is  equal  to  the  sum 
of  the  abscissa  and  co-ordinate. 

The  differential  equation  of  the  curve  will  be 

dx 
—  y  -J-  =  ^  +  3/5     or     xdy  -f-  ydx  -f  ydy  =  0. 
CLy 


GEOMETRICAL  APPLICATIONS. 


891 


.  • .  7/'^  -\-  2x1/  =  (7,  a  hyperbola. 
4.  The  curve  in  which  the  subnormal  is  constant. 

Subnormal         =  y—-  =  a.     .  • .  ydi/  =  adx. 
ax 

.  • .  y2  =  2ax  +  (7,  a  parabola. 

172.  Prop.  To  find  the  equation  of  a  trajectory  or  curve  which 
shall  intersect  all  the  curves  of  a  given  family  in  the  same  angle. 

Let  (p(a:,y,a)  r=  0 (1) 

be  the  equation  of  a  class  of  curves,  in  which  the  parameter  a  may  take 
any  value ;  and  let  t  —  tang  13,^  ,      /  B, 

where  (3   represents    the  con- 
stant angle  of  intersection. 

Suppose  a  to  take  a  parti- 
cular value,  «!  and   let  A^B-^ 
be  the  particular  curve  in  the 
general  class  resulting  from  this  supposition.     Then,  if  x^^  denote 
its  general  co-ordinates,  its  equation  will  be 

9(^1,2/1,  «i)  =  0 (2), 

and  the  differential  coefficient  ~,  derived  therefrom,  will,  when  ap- 

CiX-i 

plied  to  the  point  P,  express  the  trigonometrical  tangent  of  the  angle 
PTX  or  Vj,  included  between  the  tangent  PT  and  the  axis  of  x. 
Also,  if  X  and  y  denote  the  general  co-ordinates  of  the  required 

dif 
trajectory  CPD,  the  differential  coefficient  -^,  given  by  its  equation, 

will,  when  applied  to  P,  give  the  tangent  of  PLX  or  v. 


0    L 


But  /3  =  Vj 

or  by  substitution, 


V.     .  • .  tan  /3 


i== 


dj/i       dy 
dx^       dx 

1  -J-  ^.^ 
dxy^    dx 


tan  Vj  —  tan  v 
1  4-  tan  Vj  tan  v 


(3). 


^92  INTEGRAL  CALCULUS. 

Now  at  the  point  P,  where  the  curves  A^B-^^  and  CPD  inter.sect, 
x^  =z  X  and  yj  =  y.     Hence  -—  can  be  expressed  in  functions  of 

(IX  t 

X,  y,  and  a^,  and  thereft)re  (8)  may  be  written  thus 
n^,  y,%a,)  =  0....  (4). 
But  (2),  when  applied  to  P,  gives 

9(^,2/,  «i)  =  0 (5). 

If  then  «!  be  eliminated  between  (4)  and  (5),  the  resulting  equa, 
tion,  being  independent  of  the  position  of  the  particular  curve  A^B^, 
will  apply  to  all  points  in  the  required  curve  CPD. 

173.  1-  Determine  the  curve  which  cuts  at  right  angles  all  straight 
lines  drawn  through  a  given  point. 

Let  X2^J2  ^^  ^^®  co-ordinates  of  the  given  point. 

The  equation  of  one  of  the  straight  lines  passing  through  that 
point  will  be  of  the  form 


.  • .  9(a^i2'i«i)  -y\-y2  —  ai(-^i  -  a-g)  =  0   and     -j:^  =  a 

dy^      dy 


dx,  ~  '~^* 


Also     ^  =  tan  -  -r  =  00.     .  • .  — —- =  od.     and  consequently 

dx^    dx 

Also  at  the  point  of  intersection 

y-y%-  «i(^  -  .Ta)  =  0 (2). 

Now  eliminating  a^  between  (1)  and  (2),  we  get 

dy 
*  —  ^2  +  ^  (y  —  2^2)  =  0,     or    (tdx  —  x^dx  -f  ydy  —  y^dy  =  0, 

which  is  the  differential  equation  of  the  curve. 


GEOMEPRICAL   APPLICATIONS. 


*ms 


And  by  integration  ^  a:^  —  3C2X  +  7,  y^  —  3/2^  =  ^> 

or  {x  -  x^f  +  (y  -  ^2)'  =  2C  +  ar^^  +  y^^. 

Hence  the  curve  soiiglit  is  a  circle  whose  centre  is  at  the  point  x^,  y2> 

and  the  radius  arbitrary. 

2.  The  curve  which  cuts  at  an  angle  of  45°  all  straight  lines 
drawn  through  the  origin. 

Here  <p{x^,  y^,  a{)  =y^  —  a^x^  =  0. 

dx 


flj     also     t  =  tan  45°  =  1. 


1    I        ^'J 


(I),     and    y  —  a^x  —  0 (2). 


Eliminating  a^,    1  +  -  .  —  =  ' -—     or    xdx  +  ydy  =  yc?^  —  xdy. 

This  being  a  homogeneous  equation,  it  will  be  rendered  exact  by 
multiplying  by   ^—  =  ^,. 

xdx  -f-  yrfy  _  ydx  —  a*o?y        ^  Voi?  -f  yH  i 

•  •  "^^  L  c^  J 


a:-*  -I- 


x^  +  y^ 


tan-i?'. 
x 


Put  y  =  r  sin  ^,  a;  =  r  cos  {^  —^')—  —r  cos  ^.    .-.?•=  {x^  -f  ?/2y 


and 


tan 


• .  log  -  z=  &     and     r  =  ce  , 


the  equation  of  the  logarithmic  spiral. 

3.  The  curve  which  cuts  at  right  angles 
all  parabolas  having  a  common  vertex  and 
coincident  axes. 

Here   (^{x^,  y^,  a^)  =  y^^  —  2a ^x^  =  0. 


*  dx^ 


y 


Also  t  =  00. 


.  1  +  ^-1  =  0.... (1), 


894                                    INTEGRAL    CALCULUS, 
and  y'^  —  "Xa^x  =  0 (2). 

Eliminating  Cj,     1  +  ~-'-~  =  0    or     2xdx  +  ydy  =  0. 

tiX     CLX 

••-'  +  2^^=''^'    or    ^  +  |,  =  I. 
This  is  the  equation  of  an  ellipse  whose  axes  have  the  ratio  1 :  y/S! 


CHAPTER    III. 


DIFFERENTIAL    EQUATIONS    OF    THE    FIRST    ORDER    AND    OF   THE 
HIGHER    DEGREES. 

174.  These  equations  contain  the  several  powers  of  the  coefficient 

dy 

-~  to  the  n^^  power  inclusive  where  n  denotes  the  degree  of  the 

equation.     The  most  general  form  of  such  an  equation  is 

g+-|S+«^S  +  - +  .|+.=  o....(i). 

which  equation  can  be  derived  from  its  primitive  only  in  attempting 
to  eliminate  the  n**  power  of  a  constant  c  between  the  primitive 
and  its  direct  differential.     For  the  direct  differential  contains  only 

the  first  power  of—,  and  therefore  cannot  be  identical  with  (1)  ;  but 

if  we  suppose  the  primitive  to  contain  several  powers  of  the  same  con- 
stant c,  as  c',  c^,  c^  •  •  •  •  c",  .and  resolve  with  respect  to  c,  there  will 
result  n  values  of  c,  from  each  of  which  c  will  disappear  by  differen- 
tiation  ;  and  each  of  the  resulting  differential  equations  will  contain 

d/u 
only  the  first  power  of -^,  each  being  a  factor  of  (1 ).   Hence  by  mul  iiply- 


DIFFERENTIAL  EQUATIONS.  395 

mg  together  these  n  equations,  we  shall  produce  (1).     If  therefore 

dv 
we  resolve  (1)  with  respect  to  —^  thereby  ascertaining  its  n  con- 
stituent factors  of  the  first  degree,  then  integrate  each,  annexing  the 
same  constant  c  to  every  result,  and  finally  multiply  the  results  to- 
gether, the  complete  primitive,  which  includes  all  these  separate 
results,  will  be  obtained.  It  will  be  obvious,  that  in  order  to  render 
this  method  applicable  to  all  equations  of  the  first  order,  it  would 
be  necessary  to  have  a  process  for  the  solution  of  equations  of  all 
degrees. 

Unfortunately  no  such  process  is  known. 

175.  1.  To  find  the  complete  primitive  of  the  equation 

.      I^  =  - w- 

Resolving  with  respect  to  —j—^  we  get 

!=+<.. ...(2),     and     !=-«.. ..(3). 

Integrating  (2)  and  (3),  and  annexing  the  same  constant  to  each, 
we  have 

y  =1  ax  -^  c  .  .  .  ,  (4),     and     y  zzz  —  ax  -{-  c  .  .  .  .  (5), 
either  of  which  satisfies  the  given  equation  (1).     It  is  also  satisfied 
by  their  product. 

{y  —  ax  —  c)  {y  +  ax  —  c)  z=z  0,      > 

or  y2  _  2cy  —  a^x^  -f  c2  =  0 (6). 

For,  by  differentiating  (6), 

2ydy--2cdy—2a'^xdx=0^  and  c=y— — • 

dx 

This  value,  substituted  in  (6),  gives 

dx  dx  dx^ 


306  INTEGRAL  CALCULUS. 

or,  by  reduction,  — ^  =  a^, 

which  is  identical  with  (1). 

f  =+«*>,    and     ^i=-aK\ 
ax  dx 

2   1-8-  '^   i  4 

.  • .  By  integration    y  =  -  «  •^■"'  +  <',  a^d  y  ==  —  n  (^  -^    +  <?. 

.  ••  (y  - 1 A^  -  <•)  (,'/  +  1'^^^^-  <^)  =  0. 

4 

(y  ~  0^  =  o  '^"^^  *^^  complete  primitive  of  (1), 

.l-:+..|-.=o....o). 

^  33  _  ?  ±  (y!+_^)*        .     ^^  ^  ^  xdx^ydy^ 

^^         y  y       '    '  '  ^      ~  y^+^* 

.  •.  a;  =  +  (^2  +  y2)  +c,  and  a;  =—(0:2+^2)^  +  c. 


or 


,..  (a;  — c—  ■y/x'^  +  y2)  (a:-c+  V^-2  +  //2^  =  0,  or  ifz=c'^~2cx. 
4.  Determine  the  equation  of  the  curve  which  has  the  property 

5  —  t/a:  4-  hy. 

fl^  ,    _2a6      dy  _\—a? 


c?.y  fffi  -y/a2  -I-  62  _  1 

5:;-r:r6i^— 6^-1 —  =  '«±^. 

.  • .  y  =  ma;  4-  «a;  4-  c,     and     y  =  wia;  —  waf  -f  <?. 


DIFFERENTIAL  EQUATIONS.  397 

.  • .  2/2  =  vfi^x^  —  n^x^  -f-  2cma:  +  c^* 

This  is  the  equation  of  two  straight  lines,  which  intersect  on  the 
axis  of  y,  and  which  become  imaginary  when  a?  -{-b"^  <C  1.     Suppose 

a  =  y/Iand  ^  -  \/|     •*•  «' +  ^' =  1'  «^  =  ^  ^^^  ^  ~  ^'=  ^* 
.  • .  7?i  =  1     and     n  =  0,     .  • .  y  =  a;  +  c, 

and  the  two  lines  become  a  single  line,  inclined  to  the  axis  of  x  in 
an  angle  of  45°. 

176.  When  the  proposed  differential  equation  cannot  be  resolved 

with  respect  to  ■—-.  its  primitive  may  still  be  found  in  certain  cases, 

the  principal  of  which  will  be  examined. 

Case  Ist.  When  the  equation  contains  only  one  of  the  variables, 
and  the  solution  with  respect  to  that  variable  is  possible. 

Let  X  be   the  variable  which   enters   into   the  equation.     Put 

dy 

-p  =  />!,  and  resolve  with  respect  to  x.   The  result  will  be  of  the  form 

x  =  (ppi (1). 

But  since  dy  =  p^dx,  an  integration  by  parts  will  give 

y  =  p^x  -  fxdp^ (2). 

Eliminating  x  between  (1)  and  (2),  we  get 

y=Pi.(pPi-  fcppi  .dp^ (3), 

in  which  the  last  term  is  integrable  as  a  function  of  a  single  variable. 
Effecting  the  integration,  we  may  unite  the  result  thus 

y  =  J^Pi (4). 

Then,  eliminating  p^  between  (1)   and  (4),  we  obtain   the  desired 
relation  between  x  and  y. 

177.  This   method   may  sometimes  be  applied   advantageously 
even  when   the  more  general   method  is  applicable,  provided  the 

differential  equation  can  be  solved  more  easily  for  x  than  for  -^« 

dx 


898  INTEGRAL   CALCULUS. 


Ex.  To  find  the  primitive  of  the  differential  equation 


^"^  "  =  r+v  =  '■^^  •  •  •  •  (^)- 


.  • .  y  =  p,^p,  -  fyp, .  dp,  =  j^^  -  fYZfJl 

=  r+77^  -  *'"'~'^^>  +  ^ ('-*)• 

But  from  (1),      p,  =  (l:Z^y,    and     1+Jt>i^  =  i 
and  these  values  reduce  (2)  to 

y={x- x^)^-  tan-i^LzL5\*+  a 

178.  When  the  equation  (still  supposed  to  contain  x  only),  cannot 
be  resolved  either  for  x  or  p^,  we  may  substitute  xz  for  p^,  and  we 
can  then  divide  every  term  by  a  power  of  x,  thereby  depressing  the 
degree  of  the  equation,  except  in  the  case  where  there  is  an  absolute 
term.  If  then  the  depressed  equation  can  be  solved  for  x  or  «,  we 
shall  have  either 


x  =  (pz, 

^y 

^1  =  ^.  ^^'P^' 

z.(pz.d{(pz), 

and     y  =  f  z.(pz.d{(pz),  .  . 

•  •  (5), 

z  =  (px, 

dy 

=  x.ox.dx, 

and     y  =  fx.cpx.dx.  .  .  . 

.(6). 

or, 


In  the  first  case  we  eliminate  z  between  (5)  and  x  =  (pz.     In  the 
second,  the  desired  relation  is  found  in  (6). 

Put    p^  =  xz,     then     a;^  4-  x^z'-^  —  ax'^z  ~  0,      and     x 


1  +  2^' 

dy  az^         ^  az^       r     az    1       a^(z^  -  22«)  ^ 

^^  =  dx  =  TTv^^  '^  =  T^^'l^+^^l  =  -T^^^^Y'^' 


DIFFERENTIAL  EQUATIONS.  899 

QZ 

This  relation,  together  with  x  =  - — — -  expresses  the  relation  be- 
;ween  x  and  y. 

179.  Case  2d.  When  the  equation  is  homogeneous  with  respect 
to  X  and  y. 

Let  n  denote  the  degree  of  the  equation  in  x  and  y,  and  put 
y  =  xz,  then  the  equation  will  be  divisible  by  x",  and  if  the  trans- 
formed  equation  can  be  solved  for  z,  we  shall  have  a  result  of  the 
form 

z  =z  9j9i,      .  • .  dz  =  d{cpp{).      But     y  =  xz^     .'.  dy  =  xdz  +  zd^^^ 
or,  dy  =z  xd((pp^)  -f  cppidx,     or,    p-^dx  =  xd((pp-i)-{-  (pp^dx^ 

^^dx^Ji^p^      and      log^^/-*^-^l  =  J.^, 

This  combined  with  y  z=  x(ppi,  gives  the  desired  relation  between 
X  and  y. 

Ex.  y  —  X2)i  =z  -/r+  2h'^  .a;. 

Put  y  =  xz,     substitute  and  divide  by  x,  then 

^  -Pi  =  VTTpi^    z  =  p,-i-  vT'+7^^  dz=dp,+-^A=, 

V  1  +Pi 
p^dx  =  dy  —  xdz  -f-  zdx  —  x\^dp^-\-  ~J^^~\-\-{'p^-^  -y/l  +  Pi^)dx^ 

^       V^+Pi'^ 

dx  dp^  V\'^P\ 


X 


Vl+y?,2        1+iV 


log  a;  =  —  log(;)i  +  -y/T+T^)—  log(l  +  P\)   +  logc. 


.    X  = 


But     y  =  xz  =. 


vT+7?(^i  +  vTT i^i^)  VI  4-  p,» 


...    ^j  =  V^ZIZ,     and     .r(c  +  y^^^y^)^  ^2^ 
the  desired  relation. 


iQQ  INTEGRAL   CALCULUS. 

180.    Case  Zd.  Let  the  form  be  .  » 

in  which    cpi    -  J  does  not  contain  x  or  y. 

By  differentiation,  ^  =  p,  =  ;,,  +  ^  ^'  +  ^?1 .  ^3, 
''  dz  dx         dp^      dx 

\        (//>!  /  dx 
This  is  satisfied  either  by  making 

.  +  ^^=0.....(^).     o.     |'=0 (.). 

Now  the  differential  coefficient  -r— ^  in  (2),  contains  onlv  p  ,  since 

dpi 

(pPi  does  not  contain  x  or  ?/,  and  therefore  (2)  contains  only  x  and 
/>!.  if  then,  we  eliminate  />j  between  (1)  and  (2),  the  rc^M  t  will  be 
a  relation  l)etween  x  and  y.  But  this  relation  cannot  bo  the  com- 
plete primitive,  because  it  contains  no  arbitrary  eo'.  -t.  We 
must  then  refer  to  the  condition  (3),  which  gives  by  int<- ration 

p^  z=z  C,  a  constant. 

It  appears  then,  that  in  the  proposed  equation,  which   s  known  asf 
ClairauWti  form,  the  complete  primitive  is  obtained  b\     imply  re* 

placing    —  by  an  arbitrary  constant. 

Ex.   1.  To  find  the  primitive  of 

(til 

Replacing  the  differential  coefficient  -■-  by  (7,  we  hnv  . 
y  -  Cx=.a{\  -¥  C^) (2). 


DIFFERENTIAL  EQUATIONS.  401 

The  correctness  of  this  solution  is  easily  verified ;  for  by  differen- 
tiating (2)  we  get 

and  by  eliminating  C  between  (2)  and  (3),  we  obtain  (1). 

2.    ydx  -  xdy  =  a{dx^  +  dy^)\  or,  y  =  x'^£+(^i  +  ^-j^a. 

Substituting    C   for  -^  we  get  y  =  (7x  -f  (1  -f  C^fa. 

181.  Case  4th.  Let    y  =z  Px -\-  Q, (1). 

when  P  and  Q  are  functions  of  p-^. 

By  differentiation,     dy  =  p^dx  =  Pdx  -f  ^dP  +  dQ. 

.*•    {Px-  P)dx-xdP  =  dQ,    and    dx -^r -p^ — dPz=z--^^. 
This  being  a  linear  equation,  its  solution  is  of  the  form 

^  =  .--^''-''.[-/A--..-^-],    or,    x  =  Fp,. 

Hence  if  jOj  be  eliminated  between  this  and  (I),  the  result  will  be 
a  relation  between  x  and  y, 

182.  1.  y=p^xip^^,.,.{\). 

dy  z=  pydx  =  p^dx  +  2xp^dp^  -f-  2p-^dpy 
.  • .  (1  —  p^)dx  —  2xdp^  =  2dp^. 
dp^  2dp-i 


,'.dx-}-2x 


Pi-^  A  -  1 


.  X  z=  e 


v  ^j  —  1 

26 


402  INTEGRAL    CALCULUS. 


1  (72 


^^i  =  1  +  ;     and  from  (1),    p^  ~     ^^ 


2.  2/  =  a  4-^i)^+.i^i2....(l). 

^iC?ar  =  (1  +  p^dx  +  art^j  +  2/?iC?p^. 

.  • .  dx  ■\-  xdp^  =  —  2pidp^,    and    x  —  e~^^^^  [—/2e-^^^^  'P\<^Pi\' 

But         €-^^^'  =  6^',     and     fe^'p,dp,  =  eP'(p^-l)  -]-C^, 

.'.  X  1^2(1 —p^)  +  Ce-Pi    where     (7=  — Cj; 


1       .    1 


and  from  (1),       p^z=  —-x  dz-  'y/4y  —  4ar  +  a:^. 
.  • .  By  eliminating  pj  we  get 


0  =  2  q=  v/42/  -  4a:  +  a:^  +  Ce 


4a4:j/4y-4z+x3 


3.  y  =  ^(;>i--v/l+i^i')- •••(!)• 

In  this  example  ^  =  0,  and  by  differentiation 


p^dx  =  p^dx  +  xdpj^  —  \l-\-~Pi'  dx  —  Pix{\  +  p^)     dp-^. 


-i. 


• .  —  =  ^^- — -^    ^        cfpi,  the  integral  of  which  is 


and  fpi  +  (c  -  a:)vT+p7  =  0 (2). 


y(c  -  a:) 


But  from  (1),  p^x  -  x^l  4- Pi^  =  y.      .  • .  i?i=  ^^>2c  -  xY 

and     vT+  ;,»  =  -  — ^— ^.   . • .;,,-  |i^-i,  =  ^^!:^- 1 
.-  .  2/2 (a;2  _  2ca;)  =  -  a;2(2f  —  xf,    or  finally    a:2  +  y2  _  2ca:. 


CHAPTER   IV. 

SINGULAR   SOLUTIONS    OF    DIFFERENTIAL   EQUATIONS. 

183.  Differential  equations  may  be  regarded  as  resulting  in  all 
cases  either  from  the  immediate  differentiation  of  their  primitives  or 
from  the  elimination  of  constants  between  the  primitives  and  their 
direct  differentials. 

184.  Taking  the  latter  and  more  common  case,  let 

be  the  complete  primitive  of  the  differential  equation 


.(.,,,|)  =  0....(2), 


where  (2)  has  arisen  from  an  elimination  of  the  constant  c  between 
(1)  and  its  immediate  differential 


[SiM],,....,,, 


Now  if  the  constant  c  were  replaced  in  (1)  and  (3)  by  any  func- 
tion of  X  and  y,  the  elimination  of  this  function  would  necessarily 
lead  to  the  same  equation  (2). 

If  then  it  be  possible  to  replace  c  by  such  a  function  of  x  and  y, 
<Jin  equation  (1)  as  shall  give  by  differentiation  a  result  entirely  simi- 
to  (3),  after  it  has  been  modified  by  a  like  substitution  of  this  fiino- 
tion  of  X  and  y  for  c  ythen  the  elimination  of  that  function  would 
necessarily  lead  to  (2)  the  proposed  differential.  Hence  equation 
(1)  with  the  value  of  c  so  replaced  may  be  properly  considered  an 


4:0^  INTEGRAL   CALCULUS. 

integral  of  (2) ;  although  it  is  essentially  ditferent  in  form  from  the 
ordinary  integral  (1),  in  which  c  is  an  ai'bitrary  constant. 

Such  a  solution  of  a  differential  equation  is  called  a  singular  solu- 
tion or  a  singular  integral^  while  the  term  particular  integral  is  ap- 
plied to  each  of  the  results  obtained  by  substituting  various  constant 
values  for  c,  in  the  general  integral. 

185.  Prop.  To  determine  the  conditions  necessary  to  render  pos- 
sible a  singular  solution  of  a  differential  equation. 

Let  the  ordinary  primitive 

F(x,y,c)  =  0 (1) 

be  differentiated  regarding  c  as  variable,  and  there  will  result 

rdF(x,y,c)-\      dF(x,y,c)    (^Jc\ 

I        dx        J"^  dc  \dx}~     ' 

and  to  render  this  equation  identical  with 

R?^]- » 

which  is  obtained  by  supposing  c  constant,  the  necessary  condition 
will  be 

Now  (a)  is  satisfied  either  by  making 

^^^  =  0 (^)-        -        (S=0.....(5). 

The  condition  (5)  gives  c  =  constant,  and  therefore  (4)  can  alone 
supply  the  suitable  variable  value  of  c. 

The  equation  (4)  may  give  several  values  of  c,  and  then  there 
will  be  as  many  singular  solutions. 

186.  It  must  be  observed  that  the  value  of  c  derived  from  (4), 
is  not  necessarily  a  function  of  x  and  y,  or  of  either :  for  if  c  be 
connected  with  x  and  g  only  by  the  signs  4-  and  — ,  those  variables 


SINUGLAR  SOLUTIONS.  405 

will  not  appear  in  (4),  and  consequently  the  values  of  c  derived 
from  (4),  will  be  constants  corresponding  to  particular  integrals,  and 
not  singular  solutions. 

187.  And  again,  the  derived  values  of  c  may  be   functions  of  x 

and  y,  and  yet  not  variable.     For  if  the  primitive  (1)  be  solved 

with   respect  to  any  constant,  as  a,  appearing  in  it,  the  result  will 

assume  the  form 

«=/(^,y,c), (6); 

and  if  by  assigning  any  particular  value  to  <r,  this  value  of  a  should 
become  either  identical  with  that  of  c  given  by  (4)  ;  or  if  the  latter 
be  a  function  of  the  former,  then  c  will  be  invariable,  and  therefor*^ 
will  not  correspond  to  a  singular  solution. 

188.  If  we  solve  the  complete  primitive  (1)  for  x  and  y  succes*- 
sively,  the  results  may  be  written  in  the  forms 

^=/(y,«) (7).       y=A{^,c) (8). 

Which  differentiated  with  respect  to  c,  give  (since  the  first  members 
do  not  contain  c) 

.iM.o,     ^)=0,     .•4:=0...(0).and|  =  0,.(.0). 

That  is,  if  the  primitive  can  be  solved  with  respect  to  x  or  _?/,  we 
may  differentiate  either  of  those  values  with  respect  to  c,  placing  the 
result  equal  to  zero.  Thus  (9)  or  (10)  may  be  employed  instead 
of  (4),  when  more  convenient,  in  obtaining  those  values  of  c  which 
give  singular  solutions. 

189.  it  may  be  observed  that  no  differential  equation  of  the  fir^t 
order  and  first  degree  can  have  a  singular  solution;  for  such  equa- 
tions have  complete  primitives  containing  only  the  first  powers  of  c, 
and  these  priftiitives,  when  differentiated  with  respect  to  c,  give  a 
result  (4)  independent  of  c,  which  result  cannot  furnish  a  value  of  e. 

190.  The  relation  connecting  the  complete  primitive  with   the 


iOB^  INTEGRAL    CALCULUS. 

singular  solution,  can  be  illustrated  geometrically.  For  the  former 
always  represents  a  series  of  curves  of  the  same  class,  in  which  c  is 
the  variable  parameter,  and  as  the  process  for  obtaining  the  equa- 
tion of  the  envelope  of  these  curves  is  identical  with  that  by  which 
we  find  the  singular  solution,  it  follows  that  this  solution  must  repre- 
sent the  envelope. 

191,  1.  Required  the  singular  solution  of  the  diiferential  equation 

ijdx  —  xdy  —  a{dx^  +  dy'^)-,     or,     y  —  xp-^  +  «(1  +  Pi")-  •  •  •  (!)• 

This  example  belongs  to  Clairault's  form,  and  therefore  the  com- 
plete integral  is 

yn:c^  +  a(l+c2)*,  ....(2). 

.  •  .   -1.-X  +  ac{\  +  c2)~*=  0,     and     c  — 


-/a^- 


This  value  substituted  in  (2)  gives 


Thus  the  general  solution  (2)  represents  a  series  of  straight  lines 
all  tangent  to  the  circle  represented  by  the  singular  solution  (3). 

2.  yp^^  +  2ar^i  —  y  =  0. 

The  general  solution  of  this  example  has  been  found  to  be  (p.  396) 

dc       y/^2  _  2cx 
.  • .  c  —  a:  =  0,     and     c  =  x. 
This  value  substituted  in  the  general  integral,  gives 

3/2  =  a;2  _  2a;2,     or    y^  _|_  ^2  _  q^  ^\^q  singular  solution. 

The  general  integral  in  this  example  represents  a  series  of  para- 
bolas which  do  not  intersect,  and  therefore  the  singular  solution  can- 
not, in  this  case,  represent  an  envelope. 


SINGULAR  SOLUTIONS.  40^ 

1  ^ 

3.         a!Pr^-yi>i +  2^  =  ^'      <>^'    y^Tv^-^-^ 0)- 

This  is  Clairault's  form,  and  therefore  the  general  solution  is 

w-c^  +  I.  h2\         '    ^±-x--^-^     and     c-^^f^* 

y-cx  +  ^,...,{2y       .    .    ^^.-^      2^2-^'     and     c_y-. 

This  value  in  (2)  gives 


y  =  y  2  ^^  "^  V  2  ^"^  "^  ^  V  2  ^^'     ^^    ^^  "^  ^^^' 

Here  the  singular  solution  represents  a  parabola  tangent  to  a 
series  of  straight  lines  represented  by  (2). 

192.  In  the  method  of  finding  the  singular  solution  of  a  differential 
equation,  just  explained  and  illustrated,  it  has  been  supposed  that  the 
general  solution  of  the  equation  was  known ;  but  when  it  is  not 
given  we  require  the  following  proposition. 

193.  Prop.  To  determine  the  conditions  by  which  singular  solu- 
tions of  differential  equations  may  be  found,  without  first  determin- 
ing their  complete  primitives. 

Let  u  =  F{x,y,c)  =  0 (1), 

be  the  complete  primitive  of  the  differential  equation, 

«2  =  J^2(^^y'>Pi)  =  0,  .  .  .  .   (2)  ;  and  suppose 
„^^P^-j^j,^^,^^^,^^^)^0 (3), 

to  be  the  direct  differential  of  (1). 

Also  let  U  =  f{x^y)  =  0  ....  (4)  be  the  singular  solution  of  (2), 

and  jr,=  [tel.]=/.(^,y)  =  0....(5), 

the  direct  differential  of  (4). 

Now,  whether  we  eliminate  c  between  (1)  and  (3),  or  eliminate  a 


408  IXTEGPvAL  CALCULUS. 

certain  function  of  x  and  y;  viz.,  the  value  of  c  (expressed  in  terms 
of  ar  and  y),  derived  from  the  condition 

dF(x,y,c)  _ 
dc        ~  "' 

between  (4)  and  (5),  the  result  will  be  (2)  or  its  equivalent. 

Let  (3)  be  solved  with  reference  to  c,  giving  a  result  of  the  form 

c  =  (p{x,y,p^) (6), 

and  let  this  value  be  substituted  in  (1) ;  we  shall  thus  have  (2)  op 
its  equivalent  under  the  form 

u  =  F(x,y,cp)  =  0 (7), 

where  cp  is  put  for  <?:(ar,  y,j9j) ;  for,  by  hypothesis,  (2)  is  the  result  of 
the  elimination  of  the  constant  c  between  (1)  and  (3). 

Now,  since  (2)  and  (7)  are  equivalent,  the  elimination  of  jt?, 
between  them  must  lead  to  an  identical  equation  in  x  and  y ;  that 
is,  an  equation,  which,  being  true  for  all  values  of  x  and  y,  does  not 
imply  a  relation  between  them. 

Let  Pi=f(^,l/) (B) 

be  the  result  obtained  by  solving  (2)  with  respect  to  p^ 

This  value  substituted  in  (7)  gives  the  identical  equation  before 
referred  to,  which  can  be  differentiated  with  respect  to  x  and  y, 
successively,  as  though  they  were  independent  variables,  since  the 
equation  does  not  imply  any  relation  or  mutual  dependence  between 
them. 

Then,  differentiating  (7),  and  observing  tjiat  (p  contains  a:,  y,  and  j»i, 
while  ^1  =zf(x,ij),  we  get 

du      du   d(p  .   du    d(p    dp^ 
dx      dcp   dx      d:p   dp^    dx 

du  .   du  d(p      du    di>    dpi 

and  3-+ :>--T--|- :r-T--TJ^  =^- 

ay      d(D   d^      d(p   dp^    dy 


SINGULAR  SOLUTIONS.  409 

dp-^  /du      du   dr)\       /dii    dp  \ 

'    '   dx  ~       \dx      dp    dx)   '    \dp    djj-^r 

dp^  /du       du    dp\        /du     dp  \ 

dij    ~        \dy       dp    tiyj   '    \dp     dp^) 

But  when  the  sohition  is  singular,  we  have  the  condition 

—  =  —  =  0      -^  =  0)     and  ^  =  00 
dp       dc  ^  '    '  dx  '  dy  * 

dx        -         ^  dy        ^ 
or  — =0   and  ■—  =0. 

dp^  dp^ 

If  j^i  be  eliminated  between  either  of  the  last  two  equations  and 
(2),  the  result  will  be  a  singular  solution,  provided  it  satisfies  (2). 
Thus  we  can  find  the  singular  solution  without  previously  fmding 
the  general  solution. 

Or,  again,  from  (2),  we  have 

V.dx  J~  dx        dy     dx       dp^  \dx        dy     dx)  ~ 

dp^  ~~       \dx        dy     dx)   '    \dx        dy     dx) ' 

and,  since  the  divisor  is  infinite,  when  the  solution  is  singular,  we 
shall  have  the  condition 

which  will  give  suitable  values  of  p^  to  be  substituted  in  (2),  in  order 
to  obtain  singular  solutions. 

194.  1.  Find  the  singular  solution  of  the  differential  equation 

%  =  ^P\   —yPi  +  b  =  0 (1), 

without  previously  finding  the  general  integral. 

Differentiating  i^g  with  respect  to^j  and  placing  the  result  equal 
to  zero,  we  get 

du^      ^  ^  y 


410  INTEGRAL    CALCULUS. 

this  substituted  in  (1)  gives  • 

-I ^  +  b  =  0,     or     y^-4bx  =  0 (2). 

^x         Zx  ^  ' 

This  equation  satisfies  (1),  as  will  be  seen  by  substituting  for 
X  and  jt?!,  their  values  derived  from  (2). 

•£-(t)"-'(t)+'=»'  "  '-"+'=»• 

an  identical  equation.     Hence  (2)  is  a  singular  solution. 

or  Mr^  —  y-\-{y  —  x)p^  +  (a  -  x)p^^  =  0 (1), 

2  =  2/  -  ^  +  2(a  -  x)p^  =  0,  .'.p,=  ^ 


dp,       '       ~    .    -V-       "^^^^--^  '    -^^-^a-x) 
This  value,  substituted  in  (1),  gives 

This  satisfies  (1),  and  is  therefore  a  singular  solution. 


CHAPTER  V. 

INTEGRATION    OF    DIFFERENTIAL    EQUATIONS    OF    THE    SECOND    ORDER. 

195.  Differential  equations  of  the  second  order,  when  presented 
in  their  most  general  form,  include 

jr,  y,  -7-     and     -r-y,  and  may  therefore  be  written 


Of  these  comparatively  few  admit  of  being  integrated,  and  there- 
fore only  such  particular  varieties  of  the  general  form  as  admit  of 
integration  or  reduction  to  a  lower  order  will  be  examined. 

196.   Case  1st.  Let  the  equation  involve  only  x  and  -7-^,  the  form 
being 

Then  resolving  the  equation,  if  possible,  with  respect  to  -p^,  we 

get 

d^y  d^y 

-4  =  F-^x  —  X.      .  • .  -j^2  ^"^  —  Xdx^  and  by  integration, 

(IX  ttX 

^  =  fXdx'=  X.  4-  Cv      -'•    ^dx  =  X.dx  +  C.dx, 
dx  dx 

and,  y  =  fX^dx  +  fC^dx=:Xr,^C^x^C^.  * 

The  'constants   C^  and  Cg  being  arbitrary. 


2 

INTEGRAL  CALCULUS. 

197.  Ex. 

<i'y       1     A           ^y       i 

-J-«.3  =  0,      or,      _=a.3. 

dx^                                    dx        4          ^ 

"Idx 

-  r*  +  aW     and      «  -   "^   +  C: 

198.   C'a6-e  2c/.  Let  the  equation  involve  only  y  and  -~|,  the  form 
being 


('.g)=«- 


Resolving  the  equation,  if  possible,  with  respect  to  — ^, 
dx^  -^ly-  ^  •        ^"t  ^^2  -  dx  -  dy     dx-^^  dy' 

.-.   l^>,2^/rc(y=r,+  C„    .-.  g  =  2F,+  (7, 

dy 
and     G?^  =  — '■ ,  and  the  variables  are  separated. 

/^  r,  +  c 

199.^..    'V4^  =  0,     or     ^=.^|.  =  J=. 

rfX^  y^t,y  dx^  dx  y/^y 

c/y2  ^^-4.  ,/^  yT 

^'^^  ^-J  =^  "^  /-       >    ^^^   "^''^"^g    ^1  =  ^  -7=- 

""^  ya  ya 

,  4/a^.  di/ 

.  • .    dx  =z "^         •    ■     » 

To  integrate  this,  put     \/y+  y/b  —  z,     .  • .    y  z±{z—  -y/by, 
dy=.2(z^^)dz,    and      .  ^   rf  2;  =  i^^lZL.^^ 


DIFFERENTIAL  EQUATIONS.  413 

200.   Case  Sd.  Let  the  equation  involve  ~    and    ~     ^^^Ji  ^^^ 

Cl'Xi  QiX 


form  being 


d"Xi 
Resolving  with  respect  to  -r-^  if  possible,  we  have 

This  is  an  equation  of  the  first  order,  which  being  integrated  gives 
X  =  F^'p^,     and     y  =  fp^dx  =  fp^  -f-i  =  F^p^. 
Hence,  by  eliminating  j9„  we  obtain  a  relation  between  x  and  y, 


and  y  = ? ,  +  C,. 

Hence,  by  eliminating  p^^  we  get 

202.   Case  Aith.  Let  the  equation  involve 

«,  —    and    —  only,  being  of  the  form 


^T'^'     d^)  =  ^' 


414  INTEGRAL   CALCULUS. 

Keplacing  -j-  and  -^  by/*^    and    -^,   the  proposed   equatiofl 
reduces  to 


^(.,^.f)  =  0....(l), 


which  is  of  the  first  order  between  x  and  j^j,  and  must  therefore  be 
resolved,  if  possible,  by  some  one  of  the  methods  applicable  to  such 
equations. 

Thus,  if  the  equation  (1)  can  be  solved  with  respect  to  a?,  giving 

^^F^n (2), 

we  shall  have,  since     y  =  fpidx  =  p^x  —  fxdp^, 

y  =Pi^  ~  f^iPidPi (3) ; 

and,  by  eliminating  p^  between  (2)  and  (3),  the  desired  relation 
between  x  and  y  will  be  obtained. 

Or,  again,  if  (1)  can  be  solved  for^j  giving 

Pi  =  J'i^ (4), 

then  y  =  fp^dx  =.  JF^x .  dx^  the  integral  sought. 

If  neither  of  these  suppositions  be  true,  we  can  only  resort  to 
some  one  of  the  expedients  exhibited  in  the  foregoing  chapters. 


^03.^^.  i-g  +  ^  +  I^O- 

df) 
By  substitution  -z--  o^ 


•nd  by  integration 


^.         =C/-^=*!^     when     (7=5- 
J_ o* a*  -  (b^  -  x^y  ^ 


DIFFEKENTIAL  EQUATIONS.  415 


dy  h^  —  x^ 


dx 


^a^  -_  (62  -  x^Y 


.•,  y  =z   I  —  ,  the  desii 


desired  relation. 

.y/oT-  {h'^-x^Y 

dy  d^y 

204.    Case  6th.  Let  the  equation  involve  y,  -r-,  and    -=-j  only, 

the  form  being 

By  a  substitution  similar  to  that  adopted  in  the  last  case,  we  have 

But  -p^  = -p- • -T^  =  Pi  -p-,  and  by  substitution 

dx         dy    dx         ^  dy 

which  is  an  equation  of  the  first  order  between  y  and  pi. 

dn) 
By  substitution  — ^  —  V  ~  ''nf^Vi    —  ^' 

CLX 

•  '~dl-~d^  Tx'  ^^^     y-^i'i, 

cr  by  making  p-^  =  2z,  and  consequently  Pidpy^  =  df^, 

-T 2mz  =  y,     c?^  —  2mzdy  =  ydy. 

This  is  a  linear  equation  of  the  first  order  and  first  degree,  and 
therefore  integrable. 

206 »    Case  6th.  If  we  reckon  (as  usual)  x  or  y  as  of  the  dimen- 

dij  d^v 

sion  1,  and  agree  to  reckon  -f-  of  the  dimension  0,  and   -r-r  of  the 

dx  '  dx^ 


416  INTEGRAL  CALCULUS. 

dimension  —  1,  then  every  equation  of  the  second  order  which,  upon 
this  supposition,  is  homogeneous,  may  be  reduced  to  an  equation  of 

th«  first  order,  by  making  y  z=z  vx  and  -j-r-  ==  — 
°  dx^       X 

For,  if  n  denote  the  degree  of  the  coefficients,  the  terms  contain- 
ing  — -■—  must  have  a  factor  of  the  degree  n  -{-  \,  and  those  contain- 
ing -J-  must  have  factors  of  the  degrees.  Hence  after  substituting 
the  assumed  values  of  y  and  -r-^-,  every  term  of  the  equation  will 

O/X 

necessarily  be  divisible  by  ic",  and  thus  x  will  disappear,  leaving  an 
equation  between  v,  ^,  and  p^,  of  the  general  form 

T.  7  T  7  T  dx  dv 

But  dy  =  p^dx  =  vdx  +  xdv.     .' .  —  = 


X       Pi  —  V 

dp^       z  dx       dp^ 

Also        —"  =  —      .  • .  —  =  -^-     .  • .  zdv  =  [p.  —  v)dp,^ 
dx       X  X  z 

or  by  substituting  the  value  of  2,  obtained  by  resolving  (1),  an  equa- 
tion of  the  first  order  will  arise  between  v  and  p^^  from  which  py^ 
may  be  found  in  terms  of  v.     Then  by  eliminating  p^  from  the 

equation 

dx  dv 

X    ~  Pi  —  V 

and  integrating,  we  shall  get  log  x  =  (pv. 

Lastly,  eliminating  v  between  this  result  and  y  =  vx,  the  desired 
relation  between  x  and  y  will  be  obtained. 

207.  £:-c,  a:2  ^  -  a:^  -  3y  =  0. 

Making  —^  ==  -    and     y  =z  vx     we  get 

dx^        X 

xz  —  ;r/>j  —  ovx  =  0     or     2  — -  ^j  •—  3v  =  0. 


DIFFERENTIAL  EQUATIONS.  417 

.  * ,  z  =  2^1  +  Sv    and     (^^  +  Sv)dv  =  (^j  —  v)dpi, 
p^dv  +  vdp^  =  Pidp^  —  Svdv. 

1  3 

C  +  PiV  —  j:p^  —  ^  ^'^     or    jp^  —  2p-^v  -f  «^^  =  4^^  -f  2C7. 


7?i  —  V  =  -y/4t;2  _j_  26".     Hence 
.  ■ .  »2  =  C,  (2»  +  v'4»2  +  2e).    But  »  =  ^ 


y  = 


a:3       2(7(7i  36.  1  ^     ^^1       X 

-7^ -— i  =  aa;3 whei    777  =  «    and    -— -*  =  ». 

4Ci        4a;  x  4Ci  2 


CHAPTER    VI. 

INTEGRATION    OF    DIFFERENTIAL    EQUATIONS    OF    THE    HIGHER    ORDERS. 

208.  The  integration  of  differential  equations  of  an  order  higher 
than  the  second  is  attended  with  difficulties  still  greater  than  those 
which  have  been  overcome  hitherto,  and  in  consequence  the  number 
of  integrable  forms  is  very  restricted.  The  following  exhibit  a  few 
of  the  simplest  cases. 

1st.  Let  the  form  be     f(^,     ^^\  =  0. 

_,           d^-^y  ,  d^y       du  ^  ,  ,     .      . 

rut r  ~  u,     then     - —  =  -— ,     and  by  substitution 


^('"S=«' 


which  is  an  equation  of  the  first  order  between  u  and  x.     This  being 
resolved,  gives 

u  =  F^x.     .  • .  -— ~  =:  F^x    and     y  =  f*-'^F^x .  dx*-\ 

Q/X 

209.  Next  let  the  form  be 


id^     d-hj\  _ 
\dx^'     dx^-^f  ~ 


d^'^^ti  d^ij       d/^"it 

Put       - — ^  =  w,     then    -— -  =  -— -,    and  by  substitution 


an  integrable  form  of  the  second   order,  which  has  been  already 
examined. 


DIFFERENTIAL  EQUATIONS.  419 

210.  1.  Let  ^-^-l. 

Put  ^  —  ^  — ^ 

dx^  ~     '      c?a;*  ~"rfar' 

rfw  1 

.  • .  w-r-  =1,    or  dx  z=z  udu,    and     a;  =  —u^  +  C'v 

.',  uz=  ^2x-2C^     or    — |-=v^2x-2^. 


2. 


d^_d'^y 
dx^  ~  c/a:^ ' 


Put  J  =  «      then        ^  =  _. 

c?2|(  d'^u  du  du 

dx^  ~    '  dx^   dx  '   ~    dx 

.  • .  ■— -  =u^  4-  Cj     and     aa;  = 


d^^  VmH-"^ 


.•.:.  =  iog!H:J^l±Z!....(i). 

dy        ^    ^  p      udu  , ^ 

Now        p,  =  -  =  fudx=J-^====V^^:^^  +C^ 


=  «  +  ^3^  +  C,  .  .  .  .  (2). 


420  INTEGRAL  CALCULUS. 

Then,  to  eliminate  u  between  (1)  and  (2),  we  get  from  (1) 
C\e'  z=  M  4-  y^-w^  +  C\,     a/e^''  —  2u  C^e'  -}- u^  =  u^  +  C^ 

which,  substituted  in  (2),  gives  a  result  which  may  be  written  in  tht 
*brm 


CHAPTER   VII. 

INTEGRATION    OF    SIMULTANEOUS    DIFFERENTIAL    EQUATIONS. 

211.  In  the  applications  of  the  Calculus  to  Physical  Astronomy, 
it  occurs,  not  unfrequently,  that  several  variables,  as  ic,  y,  /,  <fcc. 
are  connected  by  co-existent  relations,  the  number  of  such  relations 
being  one  less  than  the  number  of  variables ;  and  the  object  pro- 
posed is,  to  deduce  equations  which  shall  express  the  values  of  ar,  y, 
dsc.  in  terms  of  the  remaining  variable  t.  The  following  solution  of 
some  of  the  simplest  cases  of  such  equations  was  first  given  by 
D'Alembert. 

212.  Prop.  To  resolve  the  system  of  equations, 

in  which  J,  jB,  C,  Z),  ^j,  B^^  Cj,  and  i>i,  are  constants,  and  T'and  7*, 
functions  of  t;  so  as  to  express  x  and  y  in  terms  of  t. 


DIFFERENTIAL   EQUATIONS.  421 

Eliminating  first  -^,  and  then  — ,   we  can  reduce  the  proposed 

eq.uations  to  the  forms 

^^ax^-by  =  T^.,,.{\),     and     ^+a,x -\-h,y  =T^  ,  .  .  {% 

in  which  T^  and  T^  are  also  functions  of  t,  and  a,  6,  a■^^  b^  are  constants. 
Multiply  (2)  by  an  undetermined  constant  m,  and  add  the  resulting 
product  to  (1). 

.  •.    A(^  +  ^y)  +  („  +  ™„J    (,  +  l^/)  =  ^^+  »^3. 

TI.T  -1  T  1  T      •  ^     +     ^^^1 

Now  determme  m  by  the  condition    m  =i ; 

a  -f  7na^ 

or  /wa  -+-  m^tti  —  b  —  mh^  =  0, 

and  suppose  m^  and  ?«2  ^^  ^^  ^^®  ^^*^  values  of  m  given  by  this 
quadratic. 

Also  put     a  4-  ^i^i  =  ^1     and     a  -f-  m^a^  =  r^.     Then 


dt 


(x  +  Wjy)  +  rj  (a;  +  m^y)  =T^^  m^T^ 


3? 


—  (a:  4-  m^y)  ■\- r^{x -{■  m^jj)  =  T^-h  m^T^. 

These  being  linear  equations  of  the  first  order,  their  solutions  will  be 

x-\-,miy—e^''^*  [fe^^*  (^2"^^*i^3y^]  )  from  which  x  and  y  may  be 
x+m^=ze-^^t  [/e'-a*  (^2+^27^3)0?^]  )  found  in  terms  oft. 

213.  £!x.  Let  ^  +  4y  +  6x  =  e*,  and  -^  +  x  +  2y=z  e2*  be  the 
proposed  equations. 

As  these  have  the  forms  (1)  and  (2)  of  the  last  article,  we  mul- 
tiply  the  second  by  m,  and  add. 

.  •..    -i^{x  +  my)  4-(5  -^m)\x-\-  ^       ^^y\  =  c*  +  me^^- 


i22 

INTEGRAL  CALCULUS. 

Put 

m 

fi^'",  or  »^  +  3« 

5  +  m 
.  • .  7/ii  =  1,     and    Wg  =:  —  4,  ^-j  =  5  +  1  =  6,  rg  =  5  —  4  =  1. 

7  o 

«  -  4y=  «-•[/«<(«'-  4e2')rf<]  =  e-<  Ue'*  —  |e"  +  C, J 

from  whi<}h  x  and  y  are  readily  found. 

214.  Prop.  To  integrate  the  system  of  equations, 

~+(Ax  +  By+Cz)=T....(l). 

^+(A,x+Bj/+C,z)  =  T,....(2). 

dz 

—  +  (A,x  +  £0+C^)  =  T,....  (3), 

in  which  A^  B,  C,  &C.  are  constants,  and  T,  T1T2  functions  of  t. 
Multiply  (2)  by  w,  and  (3)  by  n,  and  add. 

.  • .  —  {x  -^  my  +  nz) -^  (^  +  A'^  +  ^2^0 

/        JB-^B,m^  B^n  C  +  fim  +  C^^    \       T  4- T  m  J.  T  ^ 

r  "^  ZTA^TA^^  +  A  +  A^^A^l  =^T+T,m  +  T,n, 

Hence,  if  we  put 

X  -\-  my  -\-  nz  z=  v,     and     A  +  A^m  -{-  A.^  =  if, 
and  determine  m  and  n  by  the  conditions 

B  +  J5im  4-  -i^jii  (7  +  C^m  4-  CjW 


Wl  =  — ~ r^— .      n  = 


-4  +  -^jW  4-  A^n  A  -\-  AfTh  -f-  ^j'* 

the  equation  will  assume  the  form 
dv 


(4), 


+  il/i;  =  r  -h  7\w  +  T^g^i,  which  is  a  linear  equation. 


DIFFERENTIAL  EQUATIONS.  428 

This,  being  integrated,  will  give  a  relation  between  v  and  t.  Also, 
in  finding  the  values  of  ?/i  and  n  from  equations  (4),  two  cubic  equa- 
tions will  arise,  and  therefore  each  of  these  quantities  will  have  three 
values.  Denoting  them  by  m-^^m^-,  and  wig,  n^^n.^^  and  Wg,  and  represent- 
ing the  three  values  of  the  second  member,  after  integration  by 
/7i,  U^t  and  C/'g,  there  will  result  three  equations  of  the  form 

X  4-  m^y  4-  n^z  =  U^ 
(P-^-m^y  -\-  n^z  =  t/g, 
from  which  a;,  y,  and  z,  can  be  found  in  terms  of  t, 
215.  Prop.  To  integrate  the  system  of  equations. 

(P'T 

-^  +  ax  +  hy  +  c  =  0. ,..{}), 

^  -h  a^a;  +  6iy  4-  ci  =  0  .  .  .  .  (2). 
Multiply  (2)  by  w,  and  add.     Then 


\x 

+  wiy  4-  C' )  +  (a  4-  wia 

Put 

m  =  — \ 

u  =x  -\-  my  -{-  C, 

C4-WC. 

and  C  =  — ; -,     and    a  +  ma.  =  —  w^ 

a  4-  wiaj 

and  the  equation  will  reduce  to 

The  integral  of  this  equation  is 

Hence  if  m^  and  ^2  be  the  values  of  m,  deduced  from  the  assumed 
relation  of  m  and  the  constants,  then 


424       "W  INTEGRAL  CALCULUS 

Here        — = n— ; =  w.     .  • .  m^  +  5w  =  —  4. 

^ .,  ^j  _  _  1^     and    ^2  =  —  4.     .  • .  Wj  =  2    and    Wj  =  V^' 

.  •.  «  =  i  4-  4(7^62*  +  4C6^2t  _  (7^e»v^  _  Cse'*y/'\ 


CALCULUS  OF  YAEIATIONS. 


CHAPTER   I. 


FIRST    PRINCIPLES. 


1.  Tn  the  general  expression  u  =z  9  (a^i,  iP2?  ^3  •  •  •  •  ^n)?  which  signi- 
fies that  t*  is  a  function  of  several  independent  variables  x^,  ^Tg,  ^^..x^, 
the  value  of  u  obviously  depends  upon  two  essentially  different  con- 
siderations, viz. :     1st.  The  values  of  the  variables  x-^,  ^25  ^3 ^»? 

and  2d.,  the  form  of  the  function  cp. 

2.  The  consideration  of  the  changes  imparted  to  u  by  changes  in 
the  values  of  the  independent  variables,  while  the  function  <p  is  sup- 
posed to  retain  the  same  form,  is  the  chief  object  of  the  Differential 
Calculus,  and  then  the  form  of  the  function  is  supposed  to  be 
known.  But  there  are  many  cases,  especially  in  questions  relating 
to  maxima  and  minima,  in  which  the  form  of  the  function  necessary 
to  fulfil  some  specified  condition,  is  the  principal  object  of  inquiry. 
For  the  resolution  of  such  questions,  the  ordinary  methods  of  the 
Differential  Calculus  do  not  suffice,  and  their  consideration  is 
reserved  for  the   Calculus  of  Variations. 

3.  There  are,  it  is  true,  some  cases  in  which  it  becomes  necessary 
to  consider  the  change  in  u  due  to  both  these  causes,  namely,  a  change 
in  the  values  of  the  independent  variables,  and  a  change  in  the  form 


4:26  CALCULUS  OF  VARIATIONS. 

of  the  function,  but  it  is  with  the  latter  that  the  Calculus  of  Variations 
is  more  immediately  concerned. 

4.  The  form  of  a  function  may  be  so  connected  with  the  form  or 
forms  of  one  or  more  other  functions,  that  when  the  latter  are  given, 
the  former  will  become  known.  For  example,  a  differential  coefficient 
has  a  certain  form  always  deducible  from  that  of  the  function  itself. 
This  connection  between  functions  is  expressed  by  calling  the  original 
function,  whose  form  is  arbitrary,  the  primitive,  and  that  whose  form 
is  dependent  upon  it,  the  derived  function. 

Now  if  the  form  of  one  or  more  of  the  primitive  functions  be 
supposed  to  change,  the  form  of  the  derived  function  will  undergo  a 
corresponding  change,  and  if  the  relation  connecting  the  forms  of 
the  primitive  and  derived  functions  be  invariable,  the  change  in  the 
form  of  the  latter  will  not  be  arbitrary,  but  will  be  connected  with 
the  change  in  the  form  of  the  former  by  a  fixed  relation. 

5.  To  trace  this  dependence,  or  to  investigate  the  change  in  a 
derived  function  resulting  from  an  arbitrary  change  in  the  form  of  its 
primitive^  is  the  design  of  the  Calculus  of  Variations. 

6.  In  this,  as  in  the  Differential  Calculus,  it  is  usually  necessary 
that  the  increments  of  the  function  shall  admit  of  being  indefinitely 
diminished,  and  also  that  such  increments  shall  continue  indefinitely 
small,  when  any  values,  consistent  with  the  conditions  of  the  ques- 
tion, are  assigned  to  the  variables  ar^,  a^gj  <Szc. 

Hence  the  necessity  of  the  following  proposition. 

7.  Prop.  To  investigate  a  general  "method  of  giving  to  a  function 
such  a  change  of  form  as  shall  impart  to  it  an  increment  of  any  pro- 
posed order  of  m.-ignitude,  without  reference  to  the  values  of  the 
independent  variables  arj,  a^g,  x^. .  .  .x^  which  enter  into  it. 

Let     u  =  9(^1,  X2,x^ x^)  be  the  original  function,  and 

^  Mj  z=z  (pi(j*i, x^^x^.... ar„),  after  it  has  undergone  the  required 

change  of  form  ;  and  suppose  i  to  represent  a  small  quantity  of  the 
same  order  of  magnitude  as  that  which  we  desire  to  impart  to  the 


•       FIRST  PRINCIPLES.  427 

difference  u^  —  u,  so  that  if  u^  —  u  =  ni,  the  quantity  n  shall  be 
neither  excessively  great  or  extremely  small.     Then 

i       ~~  i 

musi  be  finite  for  all  values  of  x^^  a-g,  x^..  .  x^^  consistent  with  the 
conditions  of  the  question.     Assume 

i 

u^  —  u=zi.\{x^,x^,x^,,.x^     or     u^  =  u -\- i.-\^{x^,X2,x.^. .  .x^)', 

in  which  the  function  4^  is  subjected  to  no  condition  but  that 
of  not  becoming  infinite  for  any  values  of  arj,  iTg,  x^. .  .x^  within  the 
restriction  of  the  problem. 

Hence,  in  order  to  impart  to  a  given  primitive  function  such  a 
change  of  form  as  shall  cause  it  to  receive  an  increment  susceptible 
of  indefinite  diminution,  we  must  add  to  it  another  arbitrary  function 
of  the  variables  (subject  to  the  above  restriction),  multiplied  by  a 
constant  e,  which  constant  is  to  be  assumed  of  the  same  order  of 
magnitude  as  that  proposed  to  be  given  to  the  increment  of  the 
function. 

8.  Ex.  Suppose  u  =  sin  x,  where  x  can  take  any  value  between 
0  and  -TT,  and  let  the  increment  Wj  —  u,  proposed  to  be  given  to  n  by 
a  change  of  form,  be  required  of  the  same  order  of  magnitude 
with  dx. 

Then  making  i  =  adx,  when  a  is  nearly  equal  to  unity,  we  may 
write 
Ui  =  u -\-  i  cos  X,    or    u^^iu  -\-  i  sin  2x,    or    ?^j  =  w  -f  i  sin  4x,  &c. ; 

but  it  would  not  be  admissible  to  assume 

u^  z=  u  -^  i  tan  a?,  .• 

because  tan  x  would  become  infinite  for  one  of  the  admissible  values 


428  CALCULUS   OF  VARIATIONS. 

of  .r,  viz.,  2.'  =r  -  "TT,  and  therefore  i  tun  x  would  not  be  necessarily 

small,  as  required. 

If  the  iiicreuient  required  to  be  given  to  u^  were  of  *he  same  order 
with  dx"^  or  dx^,  then  we  would  make 

i  =  a.  dx"^         or         i  —  a  .  dx^. 

9.  The  indefinitely  small  change  in  the  value  of  a  function  pro- 
duced by  a  change  in  its  form,  is  called  a  variation,  and  it  appears 
that  the,  variation  of  a  primitive  function  is  entirely  arbitrary,  but 
the  variation  of  a  derived  function  is  dependent  upon  that  of  its 
primitive,  and  therefore  not  arhitrar\. 

10.  Prop.  Let  u  =  y(x-^,  X2,  x^  .  .  .  .  x„)  be  an  indeterminate  func- 
tion of  x^,X2.X2 ....  .r„,  and  let  v  =.  Fu  denote  a  relati<jn  by  which  v 
is  derived  from  u,  that  is,  a  rehition  i)^  form,  but  not  oj  magnitude : 
it  is  proposed  to  find  the  change  in  the  value  of  the  derived  function 
(or  the  variation  of  v)  resulting  from  an  indefinitely  small  change  in 
the  form  of  w. 

Let  (p(.ri,d:2%  ....  ^„)  be  replaced  by 

^{x-^.x^.x^ x^)  -f  i .  -vl  (a;^,  x^.x^ x^\ 

and  let  the  operation  denoted  by  the  sym  lol  i^' be  performed  on  the 
substituted  function  so  far  as  to  obtain  the  coefficient  of  the  first 
power  of  i  in  the  development  of 

If  the  co-efficient  of  this  term  be  denoted  by  w,  then  will  t .  w  be 
the  variation  of  v.  This  will  appear  by  reasoning  entirely  similar 
to  that  employed  in  the  Diflferential  Calculus,  in  finding  the  differen- 
tial of  a  function  (p.  18). 

11.  The  proposition  enunciated  above  is  far  more  general  than 
that  commonly  presented  for  consideration.  Usually  the  only 
derived   functions   necessary    to   be   considered,   are   such   as    are 


FIRST  PRINCIPLES.  429 

obtained  by  the  processes  of  differentiation  and  integration,  which 
are  represented  by  the  symbols  d  and  /respectively  ;  and  for  these 
two  cases,  the  symbol  F  is  distributive^  that  is, 

F{^  +  9')  =  i^(p  +  F^', 

Thei.  to  find  the  variation  of  v  =  i^ip,  substitute  ^  •\-  i,\  ^ox  (p, 

and  since  F{^  -{-  i.\)  =  F(^  -\-  F{i.  W 

the  variation  or  increment  given 'to  F(^  will  be  F{i,\)  or  i.F\^ 
since  i  is  a  constant,  and  therefore  not  a  function  of  a?],  a-g,  &c. 

12.  Thus  far  we  have  supposed  the  function  to  receive  the  kind  of 
increment  peculiar  to  the  calculus  of  variations,  viz.,  that  due  to  a 
change  of  form  ;  but  if  the  independent  variable  be  supposed  to 
change  also,  the  function  will  receive  an  additional  increment,  and 
the  total  change  imparted  to  the  function  will  be  the  algebraic  sum 
of  the  two  increments  resulting  from  the  two  causes. 

13.  The  following  notation  is  used  to  distinguish  the  increments 
due  to  one  or  both  of  these  causes. 

l.s'if.  The  character  h  refers  to  the  change  in  the  value  of  the  func- 
tion  resulting  from  a  change  in  its  form. 

2c/.  The  character  d  refers  to  the  change  in  the  value  of  the  func- 
tion produced  by  changes  in  the  values  of  the  independent  variables 
ari,a-2,  &c. 

3c/.  And  the  character  D  refers  to  the  total  change  resulting  from 
both  causes. 

.  • .  if  w  be  a  determinate  function  of  several  variables,  then 
Du  ■=.  du. 

.  • .  If  w  be  an  indeterminate  function  of  invariable  quantities,  then 
Du  =  Su. 

And  if  u  be  an  indeterminate  function  of  variable  quantities,  then 
Du  =  du  -\-  6u. 

14.  Since  an  independent  variable  admits  of  both  species  of  change, 


i30 


CALCULUS  OF  VARIATIONS. 


we  might  denote  that  change  by  either  character.     Unless  the  con- 
trary is  specified  this  change  will  be  indicated  by  d. 

15.  The  distinction  between  differentiation  and  variation  admits 
of  a  simple  geometrical  illustration. 

Thus  let  2/  r=  (pa;  (1)  be  the  equation  of 
a  curve  A  CB  and  y^  =  (^-^x  (2),  that  of  a 
second  curve  A-^C-^B-^^  the  form  and  posi- 
tion of  the  second  curve  being  supposed 
to  differ  very  slightly  from  those  of  the 
first. 

Put  OD  =  x,  DD^  ^  dx,  DC  -y,  and  i>Ci  =  y^  Then  the 
change  NE  imparted  to  y  by  an  addition  DD^  —  dx  to  ar,  while  the 
point  referred  remains  on  the  same  curve  A  CB^  will  represent  dy ; 
the  change  CC-^  =  y^  —  y^  imparted  to  y  by  passing  from  C  to  a 
point  Ci  on  the  second  curve,  (while  x  remains  unchanged,)  will 
represent  8y  ;  and  the  change  NE-^  due  to  both  causes  will  represent 

Dy. 

16.  Prop.  Given  u  =f(xi, x^^x^.. ..  ar„)  a  determinate  function  of 
several  variables,  to  determine  its  total  increment. 

Since  the  form  of  the  function  is  supposed  invariable,  we  have 


Du  =  du  =—  dxi  +  J — .  dx2  -f  • . .  •  +  J—' dx^ \A\, 

UXj^  dX^  ^^n 


17.  Prop.  Given  u  =  (^(xi^a^^x^ . . . .  x^)  an  indeterminate  function 
of  several  variables,  to  determine  its  total  increment. 

Here  the  form  of  the  function  and  the  magnitudes  of  the  independ- 
ent variables  must  be  supposed  susceptible  of  change,  and  therefore, 

Du  =  du  -f-  Su. 


But         du  =  --  '  dx, -{- -r-  •  dxo -{- 
dxi  dx2 


4- 


du 

dx. 


dXn', 


and 


Su  =  ^  4.  (a?!,  X2,X2 x„).     Hence, 


APPLICATIONS  OF   GENERAL  PRINCIPLES.  431 

^         du   ,       ,    du     . 

+  —  -dxn-i-i--^  (ari,  x^,  arg . . .  iP„)  .  .  .  (5). 

18.  Prop.  Given  u  =  F-  (p  {x^,  x^,  x^^  .  .  .  .  ar„),  where  F  is  the 
symbol  of  a  derived  function  which  fulfils  the  condition  F  {(p  -{•  9') 
=i  Fcp  -{-  Fcp',  and  9  is  the  symbol  of  an  indeterminate  function,  to 
determine  the  total  increment  of  u. 

Here  Su  =  F[i"l.  {x^,  x^,  x^ a-„)], 

z=  F'S'Cp  {x-^,X2,x^,....x^). 

dx^       ^      dx^       ^  dx^      "^ 

-^  F'S'(p  {x^,a^,x^,...,x,) {C). 

19.  Prop.  Given    F  =  /  {x^,  Xr^,  2:3,  ...  .  ar„,  u^,  u^,  Wg, <), 

where  /  is  a  determinate  function  of  the  quantities  within  the  (  )  ; 
a*!,  2*2,  %  .  .  .  .  a:„  being  independent  variables,  and  w^,  Wg,  u^.  .  .  .  u^ 
indeterminate  functions  of  one  or  more  of  these  variables,  to  find  the 
total  increment  of  V. 

Here  V  varies  in  consequence  of  changes  in  the  values  of  ajj,  oi^^ 
rg,  .  .  a;„,  and  also  from  the  changes  in  the  forms  of  Wj,  u^.,  -Wg,  .  .  u^. 

Now  V  is  directly  a  function  of  iCj,  and  indirectly  a  function  of  x^ 
through  u^,  ^2,  ^^3,  .  .  .  .  u^.  Hence,  if  a^j  be  supposed  alone  variable, 
the  change  in  V  will  be 

dV ,      ,   dV    du^    ^         dV   duo    ^      ,  dV  du^  , 

-r-dx^  J^  --.—±,dx^  +  - -l.dx^  +  ..  .-—.—^^dx^'^ 

dx-^  du-^    dx^  du2    dx^  du^    dx-^      ^ ' 

and  similarly,  where  arg  alone  varies,  the  change  in  V  will  be 

\_dX2       duy^  dx^         du^   dx^       '  '  '  '        du^    dx^X 
and  the  other  variables  will  furnish  like  expressions. 


482  CALCULUS  OF  VARIATIONS. 

Now  let  the  form  of  the  function  u^  change,  ofeher  things  being  the 
same,  and  the  corresponding  change  in  F  will  be 

dV        ' 

since  V  is  a  function  of  w^,  and  the  change  produced  in  F  by  a 
change  in  u-^  depends  only  upon  the  amount  of  change  in  w^,  not  on 
the  manner  in  which  it  is  received. 

Introducing  similar  terms  for  the  variations  of  Wg?  %?  ....  '^'n-  «'ind 
adding,  the  total  change  in  V  will  be  thus  expressed 

i)r=r— +  — •^  +  — •-^+....  +  — -"^l/x 

\jlx^        du^     dx^        du2    dx^  du^    dx^  J     ^ 

,    VdV    ,   dV    du.       dV     du^    ,  .   dV    du-\  , 

^.dx^        du^    dx^       du^     dx^  du^     dx^j 

U  ((  ((  4( 

VdV^       dV  d^        dV  du^_  dV_   dvj\ 

L.dx^        dui    dx^         c/wg    d^n        *  '  *  *  du„    dx„  J 

the  quantity  in  the  last  line  being  the  variation  proper  or  6  T 

20.  Given  U  =  FV,  when  V  =  f  (x^,  x^,  2:3,  ...  .  a.'„,  ?/^,  u^ 
W3,  .  .  .  .  «„),  where/  is  a  determinate  function  of  the  quantities 
within  the  (  ),  and  i^a  derived  function  which  satisfies  the  condition 
^  (9  +  9')  —  F^^  +  Fq^',  to  find  the  increment  of  U. 

First,  let  aTj  alone  vary,  and  since  F  is  a  determinate  function  of 
a?i,  ^^2)  ^-q?  .  .  .  •  a^„,  Wj,  Wg,  W3,  ....«„,  it  follows  that  so  long  as  the 
forms  of  Wj,  Wg'  '^3,  .  .  .  .  w„  remain  unchanged,  the  quantity  V  will 
be  a  determinate  function  of  the  independent  variables  jtj,  x^ 
iCg,  .  .  .  .  a;„,  and  therefore  the  corresponding  change  in  U  will  be 

[-Jf]'^^"     "''•^^*     [^] 

denotes  the  total  differential  coefficient  of  XJ  with  respect  to  .r^. 


APPLICATIONS  OF  GENERAL  PRINCIPLES.  433 

And  similarly  when  x^  alone  varies,  the  correspondirg  change 

in    (7  is    I  - —  I  dx.y :    and   the   other   variables   will   furnish   like 
Xjixr^l     2' 

expressions. 

Now  to  find  the  change  in  TJ  due  to  a  change  in  the  form  of  w^, 

we  observe  that  the  change  in    CT,  resulting  from  a  change  of  any 

kind  in  w^,  might,  at  first,  appear  to  be  properly  expressed,  (as  in  the 

last  proposition,)  by owj.     Now  this  would  be  true  if  U  were 

properly  a  function  of  w^,  that  is,  a  quantity  whose  magnitude  is 
fixed  by  that  of  u^^ ;  but  such  is  not  the  case,  their  relation  being 
one  of  form,  not  of  magnitude;  and  therefore  the  desired  increment 

is  not  — (5"i«j.     But  although  U  is  not  a  function  of  Wj,  it  is  derived 

from  t^i,  the  form  of  TJ  being  dependent  upon  that  of  F,  which  latter 
depends  upon  the  form  of  Wj.     And  since  TJ  =  FV^  .  * .  hTJ  z=.  F6  V, 
But,  by  the  last  proposition, 

^  V  =  -z — 6uy  -f  -r-^Uo  +  &;c. 
au^  awg 

dV 
.  • .  F-r-  '  Su^,  is  the  part  of  dU  which  results  from  a  variatic^n  m 

the  form  of  u^. 

Hence,  the  entire  increment 


:i,". 


CHAPTER    II. 


APPLICATIONS  OF    GENERAL    FORMULAE    TO  FUNCTIONS  OF  ONE  VARIABLE. 

21.  Prop.  To  find  the  totsJ  increment  of  the  differential  coefficient 
-7-^,  y  being  an  indeterminate  function  of  the  single  variable  x. 

Here  the  quantity  proposed  can  vary  only  in  two  ways,  viz  :  by  a 
change  in  the  magnitude  of  the  independent  variable  a;,  and  by  a 
change  in  the  form  of  the  function  y,  the  case  corre'sponding  to  that 
of  formula  (C),  with  the  number  of  variables  reduced  to  one.  We 
therefore  estimate  the  two  changes  separately  and  add  the  results. 

Now  when  x  takes  the  increment  dx^ 

d*y  ,                    ,     ,         d^y   .   rf«+iy , 
u  =  -f-  becomes  u  -\-  du  = \-  .     ,^dx 

the  corresponding  change  in  ?*  being  -—q5jC?a;:  and    hence   the    total 
increment  of  u  will  be 

Du  =  du-\-5u  =  ^^dx  +  S^. 
c?a;"+i       ^      dx'' 

d^il 
But  the  symbol  -—^  satisfies  the  condition  i^((p  +  <p')  =  i^(p  +  ^^'^ 

CLX 

and  therefore 

^d*y  _  d^{y-\-Sy)       d*y  _  d*y      d^Sy      d*y      d*Sy 
dx*  ~~       dx^  dx^  ~~  dx*       dx*       dx*  ~  c?a;* 

dx""       dx""-^^    ^        dx*-* 


FUNCTIONS   OF  ONE  VARIABLE.  485 

22.  It  IS  to  be  observed  that  6y  requires  a  certain  restriction ; 
for  it  was  shown  that  when 

it  is  necessary  to  assume  the  function  4^  of  such  form  as  not  to 
become  infinite  for  any  values  of  iCj,  iCg  &;c.,  within  the  limits  of  the 
question.  This  condition  is  sufficient  when  we  consider  only  the 
primitive  function ;  but  when  it  is  necessary  to  take  account  of  a 
function  derived  from  the  primitive,  it  becomes  also  necessary  that 
the  function  similarly  derived  from  ■\>  should  not  become  infinite  for 
any  admissible  values  of  the  variables. 

Thus  when  we  say  that  6F(^  =  iF-^^,  it  is  to  be  understood  that 
F-<\^  remains  finite  for  all  suitable  values  of  iCj,  a^g,  &;c.  In  the  present 
example,  there  being  but  one  variable  x,  we  have 

oy  z=  t .  -Lx  •  0- —  =  I .  — - — . 

d^-l/X 
and  we  must  so  select  -1  that     ,       shall  be  finite  for  all  admissible 

values  of  X. 

23.  Prop.  To  find  the  total  increment  of 


=/L^,y 


d^  ^ 
'  dx'  dx'^ 


where  y  is  an  indeterminate  function  of  x. 

This  is  a  particular  case  of  the  general  investigation  which  resulted 
in  the  formula  [/>].  To  make  that  formula  applicable  to  the  present 
case,  we  reduce  the  number  of  variables  to  one,  and  put 

dy  d'^y  „ 

Making  the  substitutions,  and  putting,  for  brevity, 

^JL-M      ^-i^      ^-P  ^-P  ^-P 

dx-'"^^       dy-'^r      Jy-"^'  .d^y~^^-----dny-^' 

d-j-  d^—  d—^ 

ax  dx^  dx* 

we  get 


486  CALCULUS  OF  VARIATIONS. 

or  by  substituting  for         d  — ,  5  -— ■  &c. 
their  values  given  by  the  last  proposition, 

24.  Here  ^y  is  to  be  expressed  as  hitherto  by  i  .■\>x^  and  therefore 
v}^  is  to  be  assumed  of  such  form  that  neither  it,  nor  any  of  its  first 
n  differential  coefficients  shall  become  infinite  for  any  value  of  x  con- 
sistent with  the  conditions  of  the  problem. 

25.  Prop.  To  find  the  total  increment  of  U  =  /    ^  Vdx      when 

^-^L'^'c^x'      dx^ dx-S 

It  is  obvious  that  a  definite  integral  can  change  its  value  only  in 
three  ways,  viz.  ; 

1st.  By  a  change  of  the  superior  limit  Xi,  while  the  inferior  limit 
Xq  and  the  form  of  the  diflferential  coefficient  V  remain  the  same ;  S'i. 
By  a  change  in  the  lower  limit  Xq,  while  the  superior  limit  and  the 
form  of  V  ai  e  unchanged ;  and  Sd.  By  a  change  in  the  form  of  V 
while  the  limits  are  invariable. 

The  complete  variation  or  total  increment  is  the  algebraic  sum  of 
the  three  separate  changes  thus  produced.  Denote  by  F,  the  value 
of  Fwhen  x  =.  x-^^  and  suppose  x-^  to  take  an  increment  dx^  Then 
Vydx-^  will  be  the  corresponding  increment  received  by  U)  for  when 


FUNCTIONS  OF  ONE   VARIABLE.  437 

Xi  takes  an  increment,  IT,  which  consists  of  an  indefinite  number  of 
terms,  each  of  the  form  Vdx,  simply  receives  an  additional  term, 
expressed  by  V^dx^. 

And  similarly,  when  Xq  takes  an  increment  dxQ,  the  correspond- 
ing increment  of  U  will  be  —  V^dxQ,  since  U  will  thereby  be 
deprived  of  one  term  expressed  by  VQdxQ. 

•  • .  DU=  V.dx.  -  V.dx.  +  8  f^^  Vdx, 

J  Xq 
and  we  must  now  find  an  expression  for  ^  /    ^  V^x,  the  change  in  TJ 

^Xq 

due  to  a  change  in  the  form  of  V.  But  the  operation  denoted  by  the 
symbol  /    ^  satisfies  the  condition  i^((p  +  9')  =.  F^  -\-  F(^'' 

*f  Xq 

.-.  S  r^Vdx  =.  TUV  +  8V)dx  -  T'Vdx 

JXq  JXq  '  J  Xq 

^  r^vdx^  rHv.dx-r^vdx=.rHv.dx. 

J  Xq  JXq  JXq  JXq 

dv    d^ij 
Now  as  F  is  a  determined  function  of  x,  y,  ~,  -r^,  &c.,  its  form 

'^'  dx    dx^         ' 

(considered  as  a  function  of  x\  can  vary  only  by  a  change  in  the 

form  of  the  function  y. 

Hence  the  variation  of  F,  found  as  in  the  last  proposition,  is 

Now,  by  applying  the  formula  for  the  integration  by  parts  to  the 
second  member,  we  get 

in  which  [Pi^y^i  and  [Pi^yjo  represent  the  values  of  P^dy  at  the 
superior  and  inferior  limits  respectively.     Similarly 


438  CALCULUS   OF  VARIATIONS. 

Jxq      ^  dx^  L^dxA^       L^dxjQ      Jxq    dx      dx         * 

or,  by  applying  a  similar  process  to  the  last  term, 

=['-.f-S"],-[''-S-t"]. 


And  if  we  integrate  n  times  successively  the  term 

y%     r^         ^"^y      7  1  .,1  T 

Pn  .  —i-^  dx,  there  will  result 

Px.  ^  d»8y     ^         r  ^  d^-^y       dPn   d'^-'^Sy 
Now  collecting  the  coefficients  of  Sy,  -^,  &c,  we  get 

^^a-o  L  c^j;  ^  dx^      06C  . . .  -TV       ;      ^      j  j 


FUNCTIONS  OF  ONE  VARIABLE.  489 

4''.-],[f],-^-H.-[f].+- 

which  is  the  expression  required. 

26.  The  value  of  i>  /    ^  Fc?a;  found  in  the  above  propositions,  con. 

tains  three  parts  essentially  different  from  each  other,  viz. : 

1st.    The   terms    V^dx^  —  ^o^^-^o?  which   are   independent  of  the 

change  in  the  form  of  F,  but  depend  exclusively  on  the  variations  of 

the  limits. 

2d.  The  cerms  \P^  —  &;c.]i^y,  which  depend  upon  the  form  of  the 

function,  not  for  every  value  of  x ;  but  for  limiting  values  alone. 
3d.  The  termxS  within  the  sign  of  integration 

which  depend  upon  the  general  change  in  the  form  of  the  function. 

27.  The  nature  of  this  difference  becomes  more  apparent  by 
observing  that  hy  =zi.  \x.  For  it  is  plain  that  the  terms  in  the  first 
class  are  wholly  independent  of  the  form  of  the  function  -^t'.  that 
those  in  the  second  class  do  not  require  for  their  determination  a 
knowledge  of  the  ^rm  of  the  function  4^,  but  only  the  values  of  that 
function  and  its  first  n  —  \  differential  coefficients,  at  the  limits ; 
and  that  the  terms  of  the  third  class  depend  upon  the  form  of  the 
function  4-,  and  cannot  be  determined  so  long  as  that  form  remains 
arbitrary. 

28.  Proip.  To  find  the  total  increment  oi  U  =   I    ^  Vdx,  when 


440  CALCULUS  OF  VARIATIONS. 

,,      ^r         dy      d-^y         d'tj  IdiA       /d^v\ 

^  =f  b' ^' i'  sJ •  •  •  •  5]^'  ^- y- (7.).'  \ii^: ■■■'" y- 
(II'  (SI-]' 

the  quantity    V  being   supposed  to  contain  explicitly  the  limiting 

values  of  one  or  more  of  the  quantities,  a:,  y,  — ,  &c. 

Since  a^j,  y-^  and  a-^,  y^  are  connected  by  the  same  general  relation 

as  X  and  y,  the  integral    /    ^  Fc/ar  can  be  varied  only  in  the   three 

methods  explained  in  the  last  proposition. 

Now  when  x-^  receives  the  increment  dx^^  the  form  of  the  function 
y  remaining  unchanged,  the  increment  received  by  ?7  will  be 

Similarly,  when  Xq  receives  an  increment  dx^^  the  change  in  U 
(vill  be 

\dxjQ 
Now  let  the  form  of  the  function  y  change,  while  other  things 
remain  the  same,  and  the  corresponding  change  in    U  will  be 

JdSyK     rx.JV^  ^^{dSjA     r.,_dV_  _  ^^ 
\dxfi'fx„  ^/M  \dx/itJxo  ^/d£\ 

\rf2/j  \dx}„ 

^\dx^},  Jx,       IdW"  +  ^  dx^ ),  Jx,      (dhj\ 


FUNCTIONS  OF  ONE  VARIABLE.  441 

dV  dV  dV  dV 


rfF  rfF  rfF  dV 

\dx/Q  \dxyQ 

Now  integrating  by  parts,  as  in  the  last  proposition,  and  collecting 
the  terms,  we  obtain 

&c.,  <Ssc.,  <kc.,  (fee. 

29.  Pro/?.  To  find  the  total  increment  of  (T  =J    ^  Vdx,  in  which 


442 

CALCULUS 

OF  V^ 

i.RIATIONS. 

V 

=/[. 

dy  d'y 
''^'di'd?'  " 

d^y 
dx^' 

dz  d?z 
^'Tx'dx'^'' 

chz' 

d^. 

y  and  z  being  indeterminate  functions  of  x. 

dx 

=  M, 

dV              dV 

dy  -  '^'  ^dy 
dx 

=  A 

^^  -P 

'd''y'  " 

dx'' 

dV 

'  d'-y' 

dx"" 

dV 
dz 

=  iV, 

dV 

dx 

dV 
d^y- 
dz' 

—  p  '    .  .  .  . 

-^2  5 

dx"^ 

p.; 


Then,  since  the  value  of  U  can  change  only  in  four  ways,  viz. : 
1st.  By  a  change  in  the  value  of  x^ ;  2d,  by  a  change  in  the  value 
of  «o ;  3d,  by  a  change  in  the  form  of  the  function  y ;  and  4th,  by  a 
change  in  the  form  of  the  function  z ;  we  shall  obtain  by  reasoning, 
as  in  a  preceding  proposition,  where  y  was  the  only  function, 

DU=  V,dx,  -  V,dx^  +  [p,  _ ^  +  &c.]  Sy, 


•      FtJNCTIONS  OF   ONE   VARIABLE.  443 

JXq  L  ax  dx^ 

+(-i)-^]^-<'- («)• 

And  if  there  be  several  indeterminate  functions  of  x  in  the  value  of 
XJ^  each  will  introduce  a  set  of  similar  terms  in  DU  ov  5U. 

30.  Remark.  The  results  just  obtained  are  equally  true,  whether 
the  functions  a;,  y,  z,  &c.,  are  entirely  independent  of  each  other,  or 
are  connected  by  one  cr  more  equations  of  condition. 

31.  Prop.  To  find  the  total  increment  of  U  =  /    ^  Vdx,  in  which 

_      r  dy    d^y  d^'y        dz    d'^z  d^z  T 

the  functions  y  and  z  being  connected  by  the  relation  Z  =  0,  which 
relation  may,  or  may  not,  be  a  differential  equation. 

The  equation  (a)  of  the  last  proposition  is  immediately  applicable 
to  this  case,  but  since  z  and  y  are  connected  by  a  given  relation,  hz 
and  ^y  are  not  both  arbitrary,  one  being  dependent  upon  the  other. 

32.  If  the  equation  Z  =  0  can  be  resolved  with  respect  to  one  of 

the  variables  (as  0),  giving  a  result  of  the  form  z  =  Fy^  the  several 

dz    dP'Z 
differential  coefficients  -— -,  -j- ,  &;c.,  can  be  formed  by  simple  differ- 
dx    dx^ 

entiation,  and  these  values,  substituted  in  that  of  F,  will  render  it  a 

function  of  a:,  y,  and  their  differential  coefficients.    Thus,  the  case  will 

become  the  same  as  that  considered  in  a  previous  proposition. 

But  since  the  equation  Z  =  0  is  often  a  differential  equation  which 

cannot  be  integrated,  this  method  is  frequently  inapplicable.     It  will 

now  be  shown  that  by  another  method  (due  to  Lagrange)  one  of  the 

variation^!  hv  or  hz  can  be  removed  from  under  the  sign  of  integration. 


444  CALCULUS  OF  VARIATIONS. 

^  dL  dL        ^        dL  ^  dL  , 

Put      T-  =  a,     —J-  =/3,     — — -  =  7,  &c.,      -^  =r  a', 

a —  a  — 

dx  dx^ 

Now,  since  the  equation  Z  n:  0  is  true  for  all  forms  of  y  and  z  con- 
sistent with  the  conditions  of  the  question,  we  must  have  ^L  =  0. 

+  „-fe  +  ;3'f^-/^  +  &c.  =  0 (a). 

When  this  equation  can  be  integrated  so  as  to  give  a  value  of 
either  ^y  or  ^z  in  terms  of  the  other,  (as  for  example  that  of  ^z  in 

terms  of  ^y),  we  can  form  the  values  of      -^-.     -—r  k>Q,..  by  differ- 
^"  dx        dx^  '' 

entiation,  and  then  substitute  them  in  the  value  of  ^C/',  as  determined 

in  the  last  proposition,  thus  effecting  the  desired  transformation.  But 

as  this  integration  is  rarely  possible,  it  is  usually  necessary  to  adopt 

the  method  referred  to  above,  which  will  be  now  explained. 

33.  The  value  of  6V  being 

dV=my  -hF,^+  f/^  +  &c.  +  N'dz 
^  dx  dz^ 

^^'    dx  ^^^    dx^^^""' 
e  can  (without  disturbing  the  equality  nere  expressed)  add  t )  the 

second  member  of  this  equation,  the  value  of  SL  multiplied  by  an 
arbitrary  quantity  X,  since  X.dL  =  0,     Hence  we  may  write 

iF  =  (JV+  Xa)iy  +  (P,  +  X^)  J  +  (A  +  ^r)^  +  &«• 
+(iVr'+x«')&+(/','+x/3')^-+(r,'+X7')'^  +  &c. 


FUNCTIONS  OF  ONE  TAMABLE.  445 

.  •.  iU=  (P,  +\I3-  '^^^'^''''^  +  «fec.),5yi 

+  (P,  +  X,'-^c.)..(t)^ 

-(P,'  +  X/-&c.)„.(f)+&c. 

Now  let  it  be  required  to  determine  an  expression  for  SU  con- 
taining but  one  of  the  variations  Si/,  dz,  under  the  sign  of  integration. 
If  the  value  of  X  be  determined  by  the  condition 

,V'  +  Xa'-^(^M:)  +  &e.=0 
ax 

the  variation  Sz  will  disappear  from  under  the  sign  of  integration, 
and  similarly,  if  X  be  determined  by  the  condition 

ax 

8y  will  disappear  from  under  the  sign  of  integration. 

The  following  example  exhibits  an  application  of  this  method. 


446  CALCULUS  OF  VARIATIONS. 

34.  Prop,  To  fold  the  total  increment  oi  U  =  I    ^  Vdx  in   which 

--/[^-'IS g'M 

and  v-fUv'^l-'^^         .    ^1 

rfar  '  dy  '  ^dy_  ^d^y 

dx  dx^ 

dv  dv  dv  dv  o       C    i  ^^       ht, 

dx  dx^ 

The  equation  L  =  0  becomes  in  this  case 

V —  —  0         since        /  vdx  =  z.     Hence 


-—  =  --     or     a  =  w,     and  similarly     (3  =  p^,  y  =  jOg?  <^^*- 
ay       dy 

A-lso  -r-  =  — r-  or  a'  =  0  and  similarly  (3'  =  —  1,  7'=  0  &c 

a2        as 

And  by  substituting  these  values  in  the  formula  of  the  last  pro- 
position, we  obtain 

«{/=[/>,  +  X^.  -  ^^^^^  +  &c.], . Si,, 

+  [P,  +  Xp,  -  &C.J, .  (§)  -  [A  +  X;,,  -  &c.]„(g).+&c. 

^  (X  A  -  X„&„)  +  ^^'  [iVT'  +  ^]  5^  .  dx. 


FUNCTIONS  OF   ONE  VARIABLE.  447 

Since  P/  =  0,  P^  —  0  &c.,  there  will  be  no  terms   containing 

(?)  (?)  ^- 

By  adding    V^dx-^  —  VQdxQ  to  the  expression  for  SU  just  found,  we 
shall  obtain  the  total  increment  DU,  and  in  order  to  reduce  DU  to 
form  in  which  Si/  shall  be  the  only  variation  remaining  under  the 
sign  of  integration,  we  determine  X  by  the  condition 

which  gives  X  =  —  /  N'dx, 

Denoting  this  value  by  i  we  obtain 

DU=  r,dx,  -  V,dx,  +  [P,  +  i.p,-  ^~^  +  &o.]fy, 
+  [P,  +  ip,  -  &c.], .  (5)^  -  [A  +  ip,  -  &c.]„. (^) +&C. 

—  (i,fo,  —  ;„&„) 


CHAPTER    III. 


SUCCESSIVE  VARIATION. 


35.  Thus  far  no  condition  has  been  imposed  as  to  the  invariability 
of  form  of  the  function  4*  <^'*  ^V'  l^e  conclusions  arrived  at  are 
equally  true,  whether  that  form  be  variable  or  invariable. 

Thus  if  the  symbol  F  satisfy  the  condition 
F{^  +  ^')  =  /9  +  F^\ 
it  is  equally  true  that 

^i<9  =  FS(^  =  Fi .  +, 
whether  the  form  of  sj^  be  constant  or  variable.     But  this  condition 
ceases  to  be  immaterial  when  it  is  necessary  to  take  account  of  the 
second  variation^  that  is,  the  variation  of  the  variation.     Thus  in  the 
case  just  referred  to,  we  should  always  have 

6''F(^  =z  F^-'^  z=:  Fi6^. 
But  this,  when  the  form  of  4"  i^  supposed  invariable,  reduces  to 

^2i^9  =  FO. 

Now  FO  =  0,  since  by  the  nature  of  the  function  F,  we  have     • 

F{(p  i-0)=zF^-{-  FO 
.',  F.O  =  F{(p-^0)-  F<p  =  F(p  -  F^=zO.        .  • .  S^Fip  =  0. 

Hence  for  convenience  we  agree  that  the  variation  Su  of  any  func- 
tion 7/,  although  of  arbitrary  form,  shall  yet  preserve  that  form  inva- 
riable, so  as  in  all  cases  to  satisfy  the  condition 

d^u  =  0. 


SUCCESSIVE  VARIATION.  449 

36.  We  may  notice  here  a  striking  analogy  between  a  primitive 
function  and  an  independent  variable,  the  first  increment  of  each 
being  arbitrary,  and  the  second  equal  to  zero. 

37.  Prop.  To  find  the  second  variation  of  the  differential  coefficient 
•-—.     It  has  been  already  shown  that 


dy  _  d'^Si/  _     d'^i/  _  vT/^^"^!  —  ^^"^y  _  ^"'^^y 

dx*  ~   'dx^^     ^^       '   '       dx^  ~    L  dx^'J  ~     'dx^  ~    dx^  ' 


But  since  y  is  a  primitive  function         d'^y  =  0. 

d^S'^y       ^  -.  ,         ^„  dy 

,  • .  —j-^  =  ^j     ^^^  consequently      6^  — ^  =  0. 

ax  ax 

38.  Prop.  To  find  the  second  variation  of 

^-'^U^^'  dx' dx-J 

We  have  already  found 


6V 

dV 
=  di 

4 

dx 

dh 

dx 

+  ••••  + 

dV 
dy 
dx" 

d^Sy 
dx'' 

^V=. 

-[f'. 

'     + 

dx 

dxj 

But  %  =0,     ---  =  0,  &c. 


•••^W^^]^^^^ 


IT. 

dy  "-^J  — "^"  rtf^  ' 


and,  by  determining  the  value  of  ^  —  in  a  manner  s'milar  to  that 
in  which  8  V  was  found,  we  get 

^dV      d^V^  d^V  dSy  .       d^V   d^Sy 

0 3::  Oy  -j . .....  J . . 

dy         dy'^  .    ^dy    dx  ,    ,d"y   dx'^ 

29 


450  CALCULUS   OF   VARIATIONS. 

Similarly  J  [-^.^  =  ^^  i_^,  and 

dx  dx 

.    dV  d^V    ,      .         d^V     dSy 

dx  ^    dx  L    dxl 

&c.  (fee.  (fee. 

Henee,  by  substitution,  we  at  length  find 

^    dx  L  dxA 

39.  Prop,  To  find  the  second  variation  of  /  Vdx^  when 

It  has  been  shown  that  ^  f  Vdx  =  fd  Vdx,  and  similarly  we  get 

d^fVdx  =  8  [Sf  Vdx]  =  Sf6Vdx  =  fSWdx. 

Substituting  for  8"^  F,  its  value  found  in  the  last  proposition,  we 
obtain 

d^V      RdVT"      „     )    , 
dx. 


m 


\S1  *'■'■] 


By  similar  methods,  the  third   and   higher  variations  could  be 
deduced,  but  the  results  are  of  little  practical  value. 


CHAPTER   TV. 


MAXIMA    AND    MINIMA. 


40.  The  Calculus  of  Variations  is  applied  with  great  advantage  in 
resolving  questions  of  maxima  and  minima,  to  which  the  ordinary 
methods  of  the  Differential  Calculus  are  not  applicable. 

41.  A  maximum  value  of  a  function  is  one  which  exceeds  other 
values  of  that  function,  produced  by  infinitely  small  changes  in  any 
or  all  of  its  varying  elements. 

In  the  Differential  Calculus,  these  changes  in  the  values  of  the 
function  are  produced  by  changes  in  the  values  of  the  independent 
variables,  while  the  form  of  the  function  remains  the  same ;  but  in 
the  Calculus  of  Variations  the  changje  in  the  value  of  the  function  is 
due  to  a  change  in  its  form. 

42.  The  problem  of  maxima  and  minima,  as  resolved  in  the 
Differential  Calculus,  is  the  following: 

Given  u  =^fx^  where  x  is  an  independent  variable,  and  /  a  func- 
tion of  determinate  form,  to  find  what  values  of  x  will  render  u  <* 
maximum  or  minimum. 

In  the  Calculus  of  Variations,  the  corresponding  problem  is  this ; 

Let  9  denote  a  function  of  indeterminate  form,  and  u  =:  Fc^  o. 
function  derived  therefrom,  to  find  what  form  of  (p  will  render  u  a 
maximum  or  minimum. 

43.  The  mode  of  resolving  this  latter  problem  is  as  follows : 
Let  (p  -f-  *•  +  ^®  substituted  for  9  in  the  derived  function,  and  let 

/"  ((p  +  i .  4-)  be  developed  in  terms  of  the  ascending  powers  of  t. 


452  CALCULUS  OF  VAKTATIONS. 

Then,  by  a  course  of  reasoning,  entirely  similar  to  that  employed  in 
the  Differential  Calculus,  it  will  appear  that  when  (p  has  the  form 
proper  to  render  Fq)  a  maximum  or  minimum,  the  coefficient  of 
the  firfjj^.  power  of  i  must  reduce  to  zero,  and  that  of  the  second 
power  of  i  must  be  negative  for  a  maximum,  but  positive  for  a 
minimum.  In  other  words,  if  the  form  of  9  alone  be  supposed  to 
change,  we  must  have  5u  =  0.  But  when,  from  the  nature  of  the 
question,  both  the  form  of  9  and  the  value  of  x  are  liable  to  varia- 
tion, we  must  have 

J)u  =  0. 

44.  The  application  of  this  theory  will  now  be  explained,  observing 
that  in  the  present  state  of  this  Calculus,  the  functions  to  which  it  is 

iftpplied  are,  almost  exclusively,  those  having  the  form  of  a  definite 
integral,  such  as 

r^  vdx. 

45.  Prop.  Let  y  z=  cpx  be  an  indeterminate  function  of  a  single 
variable  x,  and  let  it  be  proposed  to  find  the  form  of  9,  which  shall 
render 

a  maximum  or  minimum,  the  symbol  /  denoting  a  determinate 
ftmction. 

Let        du  =  Mdx  ^  N^4-dx-^  P/P:rdx^  P-f^J^  +  &c. 
dx  dx^  dx^ 

Then     ,Jz,  =  ivr^y  +  P,^+P2^+<fcc. 

and  if  the  form  of  9  be  such  as  will  render  u  a  maximum  or  mini* 
mum  for  any  given  value  of  a:,  we  must  have 

^i^.-=0,     or     iV^^y  +  P^^-hP,^f +  &c.  =0. 
"•  dx  *"  dx^ 


MAXIMA   AND   MINIMA  OF  ONE  VARIABLE.  453 

This  equation  cannot  in  general  be  satisfied  without  destroying  the 
independent  character  assigned  to  the  form  of  the  function  4/  or  8y. 
For,  unless  the  coefficients  iV",  P^,  Pg?  ^^-i  ^®  separately  equal  to 
zero,  the  equation 


m,  +  i>r-^  +  p,''4!.+^o.  =  o. 


will  establish  a  relation  between  the  form  of  the  function  -^  or  5y, 
and  that  of  9  or  y,  which  is  inadmissible.  Nor  is  it  possible  in  gen- 
eral to  satisfy  the  separate  conditions  iV=  0,  Pj  =  0,  Pg  =  ^?  ^^-i 
since  each  of  these  equations  establishes  a  relation  between  x  and  ?/, 
or  in  other  words,  determines  the  form  of  y. 

Hence  unless  all  these  equations  should  concur  in  giving  the  same 
form  to  y,  they  would  contradict  each  other  :  and  since  this  concur 
rence  does  not  usually  take  place,  the  problem  does  not  ordinarily 
admit  of  a  solution. 

46.  If  in  the  last  proposition  the  value  of  u  should  contain  but  one 

du      d'^v 
of  the  quantities    y,     — ,     -— ,     &;c.,  or  if  by  the  nature  of  the  pro- 

posed  question,  the  value  of  all  but  one  of  these  be  fixed  for  each 
value  of  ar,  the  equation 

will  be  reduced  to  a  single  term,  and  can  therefore  be  satisfied. 

47.  Example.  Let  u  =flx,  y,  —J,  and  let  it  be  required  to  de- 
termine what  form  attributed  to  the  function  y  will  render  u  a  max- 
imum or  minimum,  it  being  understood  that  the  value  of  y  is  to  be 
given  for  each  value  of  x. 

In  this  case,  since  y  is  constant  for  the  same  value  of  ar,  8y  =  0, 
and  the  equation 

^^y  +  Pi^+  P2  ?7  +  &c.  =  0     reduces  to 
dx  dx^ 


454 


CALCULUS  OF  VARIATIONS. 


C. 


The  following  geometrical  application  will   render  chis  example 
more  intelligible. 

Prop.  To  determine  a  curve  such  that,  if  at  each  point  P  a  tangent 
be  drawn  and  produced  to  cut  two  ^ 
given  lines,  DC  and  i^iC^,  parallel  to 
the  axis  of  y,  the  rectangle  DC  X  D-^C-^ 
of  the  parte  intercepted  between  the 
tangent  and  the  axis  of  x  shall  be  a 
maximum  or  minimum ;  it  being  un- 
derstood that  the  curve  is  to  be  compared  only  with  such  other 
curves  as  pass  through  that  point. 

Let  0  be  the  origin,  OX  and  OJ^the  axes. 
Put  OD  z=a,   OD^  =  «!   0G  =  x,   GP  =  y. 

Then  we  shall  have 


DC=y-{x-a) 


dy_ 
dx 


dy 


dy 


and     D^C^=y+{a^-x)-^-^=y-{x-a;)-£ 


•.r=i>cxA^.-[.-(-«)|]  X  t-(-c.,)|]./(.,4) 


(JF=My  + A 


dx 


where     iv  —  — -      and    P.  =  — — • 
dy  ^dy 

dx 


8V  =  [2y+(a  +  a,-  2x)^^5y  +  [2{x  -  a,)  {^  -  a)^£ 


^y{a  +  a,-2x)] 


dSy 
dx 


But  since  it  is  proposed  that  the  curve  shall  at  each  point  be  com- 
pared with  such  curves  only  as  pass  through  the  same  point,  we 

must  have 

Sy  =  0 

and  therefore  the  condition  5V  =z  0,  which  is  necessary  for  a  maxi* 
mum  or  minimum,  becomes 


MAXIMA  AND   MINIMA   OF  ONE   VARIABLE.  455 

2  (a;  -  a)  (a;  -  aj)  ^  +  y  (a  +  «!  —  2a;)  =  0 

.     2^  —  -^ ^^  —  0 

y       X  —  a       X  —  a^~    "* 

whence  by  integration, 

21og2/  —  log  {x  —  d)—  log  (x  —  aj)  =  logc. 

or  log  (2/2)  =  log  [c  {x  —  a)(x  —  a^)] 

.  • .  2/2  _  c(.^  _  ^)  (^  _  ^^)^ 

the  quantity  c  being  an  arbitrary  constant. 

This  equation  obviously  represents  an  ellipse  or  hyperbola  accord- 
ing as  c  is  negative  or  positive. 

Passing  now  to  the  second  variation,  we  have 

^       dx  L    dxj 

and  since  in  the  present  case     V  ==flx,y,-j-\     and     6y  =  0 

w  e  shall  have  6^  V  =  -J-^^-,  •  f^l ' 

or  i.r  =  2(:r-«)(:.-«,)[^|j]' 

or  by  putting  for  (x  —  a)  (x  —  a^)    its  value     — 

c 


5= 


c  L  c/a;  J 


The  sign  of  this  quantity  is  the  same  as  that  of  c.  Hence  the 
curve  is  an  ellipse  when  F  is  a  maximum,  and  a  hyperbola  when  V 
is  a  minimum.  In  the  first  case  the  curve  lies  entirely  within  the 
lines  CD  and  C^B^ ;  and  ir  the  second  entirely  exterior  to  those 
lines. 


156  CALCULUS  OF  VARIATIONS. 

48.  Pro'p.  To  find  the  form  of  the  function  y,  ai  d  the  values  of 
the    limits    Xq   and   arj,    which    shall    render   the   definite   integral 

U  =  /    ^  Vdx  a  maximum  or  minimum,  when 


V  = 


the  character  /  denoting,  as  usual,  a  determinate  function. 
Here,  we  have 

DU=  V,dx,  -  V,dx,  +  [Pi  - ^'  +  ^^-J  ^yi 

Two  cases  may  occur  in  the  attempt  to  satisfy  this  equation,  viz. : 

1st.  The  variation  Si/,  or  the  f  )rm  of  the  function  4^,  may  be 
wholly  unrestricted  (except  by  the  general  condition  always  appli- 
cable to  this  function)  ;  or, 

2d.  It  may  be  necessary  to  assume  the  function  4^  of  such  form  as 
will  satisfy  some  given  condition  or  conditions. 

In  the  first  case,  the  object  proposed  is  to  determine  among  all 
possible  functions,  that  one  which  shall  render  u  a  maximum  or  mini- 
mum. In  the  second  case,  the  derived  function  is  required  to  belong 
to  a  particular  class,  each  individual  of  which  fulfils  certain  given 
conditions. 

Maxima  and  minima  belonging  to  the  first  of  these  divisions  axe 
called  absolyte,a.ndi  those  belonging  to  the  second  division  are  termed 
relative.     Taking  the  first  of  these  divisions,  put  for  brevity 


MAXIMA  AND  MINIMA  OP  ONE   VARIABLE.  457 

and  equation  (.4)  will  reduce  to  the  form 

«i  -  ^0  +f''^  b'Sy'dx  =  0 (B). 

This  equation  cannot  be  satisfied  so  long  as  the  form  of  Sy  or  -j/ 
remains  unrestricted,  unless  we  have  the  two  independent  conditions: 

ttj  —  «o  =  ^>     ^^^  6  =  0. 
For,  if  a^  —  a  be  not  equal  to  zero,  we  must  have 

a,  — Qq  =z  —  /    ^bSy'dx, 

J  Xq 

a  condition  manifestly  impossible,  since  the  value  of  the  definite 
integral  in  the  second  member  cannot  possibly  remain  invariable; 
while  we  are  at  liberty  to  change  arbitrarily  the  form  of  the  quan- 
tity to  be  integrated  ;  but  the  value  of  a^  —  Oq,  which  depends  only 
upon  the  values  which  certain  quantities  have  at  the  limits,  will  not 
necessarily  vary  with  a  change  in  the  form  of  5y,  Hence,  we  must 
have 

flj  —  Oq  =  0,     and      /    ^  b5y  .dx=zO. 

Now  this   last  equation  cannot   be  true  for  every  form  of  Sy, 
unless  b  =:  0,  or 


458  CALCULUS  OF  VARIATIONS. 

a  differential  equation  which  serves  to  determine  the  form  of  the 
function  y. 

49.  The  two  equations,  a^  —  a^  =:  0,  and  6  =  0,  differ  essentially 
in  their  signification,  the  latter  establishing  a  general  relation  between 
the  variables  x  and  y,  while  the  former  connects  the  particular  values 
which  these  quantities  have  at  the  limits  of  integration. 

50.  Without  this  distinction,  the  solution  of  the  problem  would 
be  impossible,  since  there  could  not  be  two  general  relations  between 
X  and  y. 

51.  The  coefficients  of  the  increments  in  the  equation  a^  —  «o  =  ^ 
being  constant,  and  the  increments  themselves  either  entirely  arbi- 
trary, or  restricted  by  a  limited  number  of  conditions,  that  equation 
will  be  equivalent  to  as  many  distinct  equations  as  can  be  formed  by 
placing  equal  to  zero  each  of  the  coefficients  of  those  increments 
which  remain  arbitrary,  after  we  have  eliminated  all  such  increments 
as  are  restricted  by  the  given  conditions.  We  now  proceed  to  show 
that  the  equations  thus  formed,  together  with  that  obtained  by 
integrating  the  differential  equation  6  =  0,  will  just  suffice  for  the 
complete  solution  of  the  problem  when  a  solution  is  possible. 

52.  The  differential  equation  6  =  0,  or 

dx         dx2  ^        ^     dx"^  ^    ' 

d  ^y 
is  in  general  of  the  2n**  order.     For   since    V  contains  — — , 

d  V  d^v 

the  quantity  P»  =  — 7^  will  usually  contain  — -^  also  ;  and  therefore 

d — ~ 
dx"^ 

—, —  will  usually  contam  -— -• 
dx""  ^  dx^"" 

Hence   the   integral   of  ((7)  will   usually   contain   2n  arbitrary 

constants. 

di]    d'^ii  d^'^'^v 

But  if  the  limiting  values  of  rr,  y,  -—,  — | T^i^  ^®  entirely 


MAXIMA  AND   MINIMA  OF  ONE  VARIABLE.  459 

unrestricted,  the  equation  a^  —  oto  =  0  will  contain  2n  -\-  2  arbitrary 
increments,  viz.  : 

in  which  case  that  equation  cannot  be  satisfied,  since  there  would  be 
formed,  by  placing  the  coefficient  of  each  arbitrary  increment  equal 
to  zero  2/1  +  2  equations,  while  there  are  but  2n  constants  whose . 
values  are  to  be  determined. 

This  result  might  have  been  anticipated,  for  it  is  evident  that  if 
the  form  of  the  function  y,  and  the  limits  of  integration  be  entirely 
unrestricted,  the  integral  may  have  any  value  from  0  to  oo ,  and, 
therefore,  cannot  admit  of  a  maximum  or  minimum. 

53.  The  nature  of  the  restriction  imposed  upon  the  limits  must 
depend  in  each  case  upon  the  conditions  of  the  proposed  problem. 

1st.  Let  the  limiting  values  of  a?,  viz.,  Xq  and  x-^  be  given  ;  that  is, 
let  it  be  proposed  to  find  such  a  form  of  the  function  3/  as  will 
render  fVdx,  when  taken  between  fixed  limits,  a  maximum  or 
minimum. 

Here  we  have  dx^  =  0,  and  dxQ  =  0,  and  the  equation  a-^—  a  =  0 
is  now  equivalent  to  the  following  separate  equations : 

[P2-&c.]i  =  0,  [P2-&c.]o=0,  &c.  &c.  &c....[P„]i=0,  [P„]o=0. 

The  number  of  these  equations  is  2n,  the  same  as  that  of  the  con- 
stants remaining  to  be  determined  ;  and  hence  the  solution  is  in  this 
case  complete. 

2d.  Let  the  limiting  values  of  both  x  and  y  be  given. 

Then  dx-^  =  0,  Sf/^  =  0,  dxQ  =  0,  Si/q  —  0,  and  the  equation 
a,  —  »(,  —  0  is  equivalent  to  2n  —  2  separate  equations,  viz. :  those 
formed  by  placing  equal  to  zero  the  coefficients  of  the  following 
inciements  : 


460  CALCDLUS  OF  VARIATIONS. 

'ET],'[g.,'B],'[g].--[£3],^E3J.- 

But  there  are  now  two  additional  equations  resulting  from  the 
substitution  of  the  given  limiting  values  of  x  and  y  in  the  general 
solution  of  the  differential  equation  6  =  0.  For  let  the  integral  of 
that  equation  be 

/Ky,Ci,C2 c^n\  =0, 

where  Cj,  rg Cgn  are  the  2/a  arbitrary  constants.     Then  we  shall 

have  the  2/i  equations 

/[•^l,  Vv  Cj.  ^2  .  .  .  .  C2„]  =  0,     f\x^,  y^,  Cj,  ^2 C2„]  =  0, 

[P3  -  &c.]i=  0,  [P3  -  &c.]o=0,  &c.  &c. . . .  [PJi  =  0,  [PJo  =  0, 
with  which  to  determine  the  2/^  constants. 

3d.  Similarly,  if  the  limiting  values  of  x^y^  and  -—were  given  the 

new  conditi<3n,  w  ould  remove  two  of  the  preceding  equations,  viz. : 


but  two  new  conditions  would  be  derived  from  the  substitution  of 

the  limiting  values  of 
the  general  solution. 


the  limiting  values  of  —  in  the  equation  obtained  by  differentiating 


/[^,y,^i,<'2 <^2n]  =  0. 

For  let  /j  [x,  y,  ^,  Cj,  Cg, c^„]  =  0 

be  the  result  of  a  differentiation  with  respect  to  x.     Then  we  shall 
have 

and  /i  [xq,  ^0,  (J^j  ,c^,c^ c^n]  =  0. 


MAXIMA  AND  MINIMA  OF   ^NE   VARIABLE.  461 

54,  Similarly,  if  the  limitirig  values  of  -7^  were  given,  two  more 

equations  would  disappear  from  the  group  obtained  by  making 
(X-  —  (?(,  =  0 ;   and,  on  the  other  hand,  two   new  equations  would 

d'^y 
result  from   the  substitution  of  the  limiting  values  of  -— ■  in  the 

equation  obtained  by  differentiating  the  general  solution  twice ;  thus 
preserving  the  total  number  of  equations  equal  to  2/i,  the  same  as 
that  of  the  constants  to  be  determined.  And,  in  general,  whatever 
may  be  the  number  of  the  quantities  having  given  limits,  the  total 
number  of  equations  will  be  2/i,  and  therefore  just  sufficient  for  the 
complete  solution  of  the  problem. 

dy 

55.  When  the  limiting  values  of  a;,  y, -^,  &c.,  are  not  absolutely 

(XX 

fixed,  but  simply  connected  by  one  or  more  equations  of  condition, 

the  variations  of  the  quantities  so  connected  are  not  independent, 

and  therefore  two  or  more  of  the  equations,  resulting  from  the  con. 

dition  ffj  —  ay  =  0,  will  be  replaced  by  a  single  equation.      Thus 

the  total  number  of  equations  deducible  from  a^  —  cxq  =  0  will  be 

diminished  ;  but,  on  the  other  hand,  a  number  of  new  equations, 

just  sufficient  to  supply  the  deficiency,  will  arise  from  the  equations 

of  condition.     To  illustrate  this,  take  the  following 

Exam-iyle.  Let  the  limiting  values  of  x  and  y  be  connected  by  the 

equations 

Vi^fi^i     and    yo=foXQ. 

The  quantities  dx^^  dy^^  dxQ,  6yQ  will  be  connected  by  the  following 
relations : 

Now,  substituting  the  values  of  ^y^  and  ^y^^  derived  from  these  equa. 
tions  in  «!  —  Oq  =  0,  and  placing  equal  to  zero  the  coefficient  of 
each  remaining  variation,  the  following  equations  will  result : 


4:62  CALCULUS  OF  VARIATIONS. 

r.^[A-f  +  .o.]^x(//.-[|])=o 

The  other  equations  being  the  same  as  heretofore. 

These  equations,  (2n  in  number,)  in  connection  with  the  four  fol- 
lowing, viz.  : 

Vl  =/l^0»       Vl  =/0^"05      /(^15  yi?  ^15  ^25  *  *  *  *  ^2«)  =^  " 
fV^O-i  Vo^  ^15  ^2'  *  *  *  '  ^2")  ^^  ^ 

will  just  suffice  for  determining  the  2n  +  4  quantities 

•^n  y^  ''^05  yo5  ^15  ^2' '  *  ■  ■  ^2«* 


.  And  if  the  limiting  values  of  x  and  —  were  also  connected  by 


dy 
r  me  Jimiimg  vaiues  oi  x  ana 

the  relations 
we  should  have 

Hence,  the  first  three  terms  in  each  of  the  quantities,  a-^  and  a^^ 
will  reduce  to  one,  and  the  number  of  equations  deducible  from 
aj  —  GTo  =  0  will  be  reduced  to  2/i  —  2»  But  we  shall  have  in  addi* 
tion  six  other  equations,  viz. :  the  four  used  in  the  preceding  case, 
and  the  two  following  : 

f  [^\^  Vi^  /I'^i,  Ci,  cg,  •  •  •  C2«]  =  0,    /'  [xq,  yo,  /o'^o,  c\,  c^r  "  ^'aj  =  ^> 
which  are  obtained  by  differentiating  the  general  solution 


MAXIMJ    iXD   MINIMA   OF  ONE   VARIABLE.  463 

/(a?,  y,  Ci,  C2, C2„)  =  0, 

dy 
and  substituting  in  the  result  the  limiting  values  of  rr,  y,  and  — . 

Thus  the  total  number  of  equations  will  be  2n  +  4,  which  is  just 
jsufRcient. 

And  the  same  result  will  be  found  true  when  the  restrictions 
imposed  upon  the  limiting  values  of  the  several  variations  are  more 
numerous. 

57.  The  exceptions  to  the  preceding  theory  will  now  be  considered. 

58.  Case  \st.  Let   F  be  a  linear  function  of  the  highest  differ- 

ential  coefficient  -7—. 

d^P 

Then  P«  will  not  contain  this  coefficient,  and  therefore     ,       cannot 

dx^ 

be  of  an  order  higher  than  2w  —  1.     Hence,  the  equation 

^_^dP,^dy_,__  d^^ 

dx   ^   dx'  -r  \        }     ^„ 

cannot  be  of  an  order  higher  than  2/i  —  1,  a^d  its  solution  will  con 
tain  2«  —  1  disposable  constants.  Thus  the  equation  aj  —  a^  =  0, 
which  is  equivalent  to  2w  equations,  cannot,  in  this  case,  be  satisfied. 

59.  It  may  even  be  proved  that  the  equation  6  =  0  cannot,  in  this 
case,  be  of  an  order  higher  than  2?i  —  2. 

For,  put  0  =  ^-        Then  F  =  ^v  +  ^', 

.  dy    d'^y  d^—^y 

where  ^  and  0'  are  functions  of  .r,  y,  — ,  yy,  •  •  •      ^J^. 


It  has  been  shown  already  that  the  equation  6  =  0  does  «iot,  in 

it  is  01 


this  case,  contain  -7-^,  and  therefore  it  is  only  necessary  to  prove 


that  it  does  not  contain  the  coefficient   ,  .     , . 

Now,  this  coefficient  cannot  occur,  unless  it  be  in  one  of  two  terms, 


464  CALCULUS   OF  VARIATIONS. 

d-^Pn-,  d-P^ 

viz:  — ; —      or       — — . 

dx''-^  dx'' 

r.  rr  .  .,  r.  dV  dV  ^  ,        d^Pn  dH 

d-^ 
dx^ 

^2n—\y  d^P 

Now  to  find  the  coefficient  of      ■,  ,    ',    in  — r— "    we  must  form  the 
vahies  of  ly),  ly^l?  *  '  *  *  \~~„)'  ^"^  reject,  m  each,  every  term 


except  that  of  the  highest  order. 

Making                         J'Ji^r  —  ^'     '^^  ^^^® 

(d&\       d8        dd    dy          dd      d'^y   ,    ^ 
\dx)-  dx"'  dy    dx""  ,^d_y_'dx-'^^^"' 

dx 

d&    du 

simi 


,      ,  d&     du       dd     d^y     .       ,  , 

Here,  the  last   term   -; —  =  —-•—-     is    the  only  term   to   be 

du    dx       du    dx" 

retained,  because  all  the  others  are  of  an  order  less  than  n.     And 

imilarly  the  only  term  in  I^t-^I  <>f  the  order  w  +  1  is 

d&    dMi_  _  d^    d^'^^y 
du   dx^  ~~  du   dx^'^^' 

y-^  I  of  the 

order  2/1  —  1  is 

d^     d^'u   __  d^     d'^^-^y 

du    dx*    ~  du    dx^"-^ ' 

Again,  since  V  =  &v-\-  &\   .'.Pn-i  =  — — r-  =  -r-  =  v~  +  —■ 

^        .a^-^V         du  du        dw 

d — 

dx*-^ 

Hence,  by  forming  the  values   of  — r^-^,  — — -^  •  •  •  •   r — r-^-, 

'     -^  ^  dx    '      dx^  dx*-^    ' 

retaining  only  the  terms  of  the  highest  order  in  each  successive  diff'~ 

entiation,  it  will  be  seen  that  the  only 'term  of  the  order  ^n  —  1 


MAXIMA  AND   MINIMA  OF  ONE  VAKIABLE.  465 

dx"^-^  du    dx^-'^       du   rfa:^"-^' 

and,  since  this  term  is  precisely  the  same  as  the  terra  of  thie  same 

d^P 
order  in  —j-—,  the  two  will  disappear  in 

d-'Pn-,  _  d^ 
dx""-^  dx""  ' 

.  •.  the  equation  i  =  0  is  not  of  an  order  higher  than  3?*Vv4.  / 

60.   Case  2d.  Let  V=:y'fx+F  (x,  p^) ,  where  Pi  =  ^'  '^ )/  » 

Here      i^=:!^=A    and  A  =  1I  =  ^M  ^  /  l 

and  since  V  is  in  this  case  a  function  of  ir,  y,  and  -^  only,  the  equa- 

dx 

tion  6  =  0  will  become  simply 

N — ^  =  0,     or,  /c  =  -j-i 

dx         '         '  ^  dx 

and  is  immediately  integrable,  giving 

P\  —  ffr '  dx  —  f^x  4-  c. 

Substituting  the  value  of  Pj,  derived  from  the  proposed  equation, 
we  shall  have  an  equation  involving  x^^  p^^  &c-,  which,  solved  with 
respect  to  jOj,  will  give  a  result  of  the  form 

Vx  =  <P(^,  c)         or         £  =  (p{x,  c) 
.-.  y  =  9i(5:,c)  +  Ci.  .  .  .  (1). 

Now  suppose  the  limiting  values  of  x  given,  those  of  y  being  in- 
determinate : 

The  equation  a,  —  a  =  0  is  then  equivalent  to  the  two  equations 
[P{\,  =  0,  and  [PJo  =  0  or  f,x,  +  c  -  0,  (2)  and  fyC^+c  =  0  (3) 

The  two  equations,  (2)  and  (3)  contain   but  one  arbitrary  constant 
c,^  and  therefore  cannot  usually  be  satisfied,  although  the  general 
30 


466  CALCULUS  OF  VARIATIONS. 

solution  (1)  contains  the  proper  number  of  constants.  Hence  the 
proposed  problem  does  not  admit  of  a  solution. 

61.  If  in  the  case  just  considered  /c  =  0.  so  that  V  =  F{x,p^) 
the  two  equations  (2)  and  (3)  become  identical,  and  the  solution  is 
then  possible:  but  it  belongs  to  the  indeterminate  class,  since  one  of 
the  constants  remains  entirely  arbitrary. 

62.  The  results  just  obtained  are  not  peculiar  to  functions  of  the 
first  order,  such  as  that  just  considered  for  if  V  be  supposed  of  such 
form  as  will  give 

and  if  the  limiting  values  of  x  only  be  given,  similar  reasoning  will 
apply.  The  equation  6  =  0  will,  in  this  instance,  as  in  the  preced- 
ing, be  immediately  integrable,  giving 

^■-^'  +  ^''-=^=^+"' 

and  the  first  two  equations  resulting  from  the  equation  «!  —  «(,  =  0, 

are  f-^x-^  -f  c  =  0,     and    f-^x^  +  c  =  0. 

These  two   equations  cannot   usually  be    satisfied  except   when 

fyX  =  0,  in  which  case  y  does  not  appear  in  the  value  of  V. 

d'v 
And  in  general  if  -7-j  be  the  lowest  differential  coefficient  appear- 

dV 
ing  in  F,  the  form  of  F  being  such  that  — rj-  =fx^,  and  if  the  lim- 

d-^ 
dx* 

ititing  values  of  x  and  of  those  coefficients  which  are  higher  than  the 

«'*  be  alone  given,  we  may  prove,  in  like  manner,  that  the  problem 

will  not  admit  of  a  solution. 

Casf  3c?.  Let  iV  —  0,  and  let  the  limiting  values  of  x  only  be 
given. 

In  this  case  the  equation  6  =  0  becomes 


MAXIMA  AND   MINIMA   OF   ONE   VARIABLE.  467 

ax         dx^  dx^ 


and  is  integrable,  giving 

dP^      d'^P^       , 

and  the  two  conditions  furnished  by  placing  equal  to  zero  the  coeffi- 
cients ^y,     and     Si/q^  viz. : 

[P,-^^  +  &c.],  =  0     and     [i>,-^+&c.]„=0 

are  equivalent  to  the  single  condition  c  =  0. 

Hence  the  equation  a^  —  gtq  =  0  is  equivalent  to  but  2^  —  1 
equations,  instead  of  2w,  and  the  problem  is  indeterminate.  This 
result  might  have  been  expected,  for  since  y  does  not  appear  in   F, 

nor  in  the  conditions  fulfilled  at  the  limits,  the  coefficient  -~-  might 

dx       ^ 

have  been  taken  as  the  principal  function,  instead  of  y,  and  then  the 

equations  given  hy  DU  —0  would  have  been  just  sufficient  to  estab- 

d'u 
lish  a  relation  between  x  and  — ,  without  arbitrary  constants,  which 

relation,  when  integrated,  must  give  an  equation  between  x  and  y, 

containing  one  arbitrary  constant. 

63.  i^i  in  the  last  case,  one  of  the  limiting  values  of  y  were  given, 

the  problem  would  again  become  determinate.      Similarly,  when 

dy 
N  =z  0  and  P.  =  0,  and  both  limiting  values  of  y  and  —  are  in- 

dx 

determinate,  the  solution  will  contain  two  arbitrary  constants,  and 

will  be  rendered  determinate  by  assigning  at  least  one  limiting  value 

to  y  and  -^• 

And  generally,  if  the  first  m  terms  of  the  equation 
,^      dP,       d'^Po      . 
dx         dx^ 


4:68  CALCULUS  OF  VARIATIONS. 

be  wanting,  and  if  there  be  no  conditions  fixing  the  limiting  values 

>(v,   —,••••.  —, — ^,  the  solutionwill  contain  m  arbitrary  constants. 
^     dx  dx""-^  '' 

The  preceding  cases  afford  the  principal  examples  of  exception  to 

the  general  theory.     We  now  return  to  the  consideration  of  that 

theory. 

64.  As  it  will  sometimes  be  possible  to  integrate  the  equation 

,,       dP,       d^P       , 

one  or  more  times  without  determining  the  form  of  the  function  F, 
and  as  the  consideration  of  these  cases  will  greatly  facilitate  the 
application  of  the  theory  to  particular  examples,  we  proceed  to 
examine  some  of  these  cases,  arranging  them  m  two  classes. 

dv  d'^xi 

65.  \it  Case.     Let  the  first  m  of  the  quantities  y,  — ,  -7-^,  &c.  be 

wanting  in  F,  or  let 

v=f\.,p pi\ 

Then  tbe  first  m  terms  of  tne  equation 

dP 

N-      ^^  +  &c.  =  0 
dx 

will  be  wanting,  and  that  equation  will  reduce  to 

dx'^  cf:c»n+i 

which  gives,^  when  integrated,  m  times, 

dP 
P« ^  +  &c.  =  Co  +  c^x  +  c^x'^  +  :  •  •  •  c^-v»i^\ 

a  differential  equation  of  the  order  2w  —  m. 

66.  Case  2d.  Let  the  independent  variable  x  be  wanting  in  F, 
or  let 


MAXIMA   AND   MINIMA  OF  ONE  VARIABLE.  4:69 

2y  d^y"} 


J     dy      d 

=^\y^dx^  d 

In  this  case,  we  have 


^-"  '"'  '  '     dx-^  dx- 


dV  =  my  +  Pid%-^  P2^3  +  ^'• 


or,  by  substituting  for  iV^,  its  value  derived  from  the  equation, 

"-'£'+ IS-" -"'•=»•"=" 

-=[-.3+l-S>4'-.S-l-S]-+- 

P„  -j-^^  dx  gives,  by  an  integration  by  parts, 
^  ^        ^  J  dx     dx"" 


470  CALCULUS  OF  VARIATIONS. 

^  dx       L    ^  dx^       dx     dx  J 
^if^dx^        dx     dx^^   dx^    dxr^^' 

^   ^"  dx-         dx     dx-^  +  ^^ +  ^      ^^  dx-^     dx         ^^^ 

which  is  a  differential  equation  of  an  order  not  higher  than  2w  —  1. 

Thus  it  appears  that  when  V  does  not  contain  the  independent 
variable  ar,  the  equation  b  =iO  can  be  reduced  at  least  one  order. 

67.  The  following  are  the  most  important  applications  of  for 
mula  (i)) : 

1st.  Let  ^=/(|) («)• 

Here   V=c-\-  P^-^hj  formula  (i)),  since  P^  =  0,  Pg  =  0,  &c. 

But  r  is  a  function  of -^-     .  • .  P,  =  — r—  is  also  a  function  of  ^— 
dx  ^  dy  dx 

di 
Hence  by  substituting  for  V  and  P^  their  values,  and  then  solving 
with  respect  to  — ,  the  result  would  take  the  form 

dy  i 

-=o,.     .■.y  =  c,x  +  c. 

Here  y  is  a  linear  function  of  x^  and  this  result  shows  that  linear 

functions  have  the  property  of  giving  a  maximum  or  minimum  value 

dy 
to  every  function  of  -j-  which  admits  of  such  a  value. 

CLX 


2d.  Let  y  =  f(y^t) W- 

Then  F  =  c  +  ^i  ^• 

dx 


MAXIMA   AND   MINIMA  OF   ONE   VARIABLE.  471 

3d.  Let  V=f(y,'^) W- 

Then  v=e+P,'^{-'j.p. 

dx^      ax    ax 

68.   Case  Sd.  Let  the  function  V  belong  at  the  same  time  to  both 

of  the  preceding  classes,  that  is,  let  the  independent  variable  x,  and 

di/      dp"  XI 
the  first  first  m  of  the  quantities  y,  -^,     -7-^,  &;c.,  be  wanting  in  F. 

The  equation  6  =  0  gives,  as  in  the  first  case  by  integration, 
dP 

Pm -^  +  &C.  =  Co  +  C^X  +  C^X'^  +  &C Cm-yX^-K 

.  •  .    Pm=  —T^  —  ^^-  +  ^0  +  <^1^  +  ^2^^  +  <^C Cm-iX"*-^, 

This  value  substituted  in 
the  differential  of  the  given  relation 

+         &c.  &c. 

r     d^:^_  d^^^;np^dr^yl 

+  Uq+c^x  +  c^x^+^zc c^_i .  a;'»-ij  ^^^  dx. 

Integrating  by  parts,  we  get 


472  CALCULUS  OF  VARIATIONS. 

^-+V    r        dr^+'^y   dp,n+^  dm^hn         .. 

+/[^6  +  c^x  +  ^2^2  4.  &c +  c^-i.T—^]'^^^^  . . . .  (JS^. 

But  since  in  general 

x'^'-j-^dx^x^  '-^-r.x^-^'- — \^r{r-X)x^-^- — ^  &c. 


4-(-l)''.r(r-  l)(r-2) 2.1 


dx 


VI— r 


if  we  put  successively  r  equal  to  (1, 2,  3, ....  wi  —  1),  and  substitute 
the  resulting  values  of  the  integrals, 

fx^-^dx       fx-^^^hx  ....  A-^— ^^r 

in   equation  (^)  it   will   be   a   flifr<^rential    equation    of   the   order 
2/1  —  m  —  1]    that  is,  the  original  differential  equation  will   have 
had  its  order  reduced  by  m  -}•  I  degrees. 
69,  Suppose  for  example  that 

Then  the  equation  6  =  0,  becomes 

dx         dx^  * 

dP 
whence  by  integration     P^  =  -y-^  -\-  c.  "^ 

and  this  value  substituted  in  the  differential  of  (1)  viz.: 


^^=[aS+aS> 


MAXIMA  AND   MINIMA  OF   ONE  VARIABLE.  473 


dx  "^     2  ^^2 


c'  +  c-^  +  P,-/. 


a  differential  equation  of  the  second  order  as  it  should  be,  since 
2/i  —  m  —  1  =  2. 

Belative  Maxima  and  Minima  of  One   Variable. 

70.  Prop.  To  determine  the  form  of  the  function  7/  =  cpx  which 
will  render  /  Vdx  (taken  between  certain  limits)  a  maximum  or  min- 
imum, when  7/  is  selected  from  those  functions  which  satisfy  the 
additional  condition  fV'dx=:c     (between  the  same  limits);    the 

dij   d  ^1/ 
quantities  V  and   V  being  functions  of  a;,  y, -^,  -— ,  &c. 

The  condition  fVdx  =  a  maximum  or  minimum,  gives 

DfVdx  =  0 (1). 

And  the  condition  fV'dx  —  c,  gives 

DfV'dx  =  0 (2). 

Multiply  (2)  by  an  arbitrary  quantity  X,  and  add  the  result  to  (1); 

then     DfVdx  +  X.DJV'dx  =  0  or  Df{  V+\V')dx  =  0 ....  (3) 

and  equation  {?))  will  include  all  the  conditions  involved  in  the  prob« 
lem,  and  will  imply  that  both  (1)  and  (2)  are  necesf^arily  true. 

For  since  by  hypothesis  X  is  an  arbitrary  quantity,  we  may  write 

I)f{V-^\Vyh  =  0     and     Df{V  -{■  \V')dx  =  0 

.  • .  Df{\  -  Xg)  Vdx  =  0     or     {\  -  \)J)fVdx  =  0. 

Now  Xj  and  X^  are  not  equal,  and  therefore  X|  —  Xg  is  not  equal 
to  zero.     Hence  we  must  have 

Of  Vdx  =r  0,         and  .  *.  from  (3)         DfVdx  =  0  also. 


4:74.  CALCULUS  OF  VARIATIONS. 

Thus  (3)  includes  all  the  conditions  required;  and  therefore  if  we 
replace  F"  by  V  +  W,  the  problem  can  be  solved  as  one  of  abso- 
lute maxima  or  minima. 

The  formula  (3)  expanded  and  applied  to  the  limits  Xq  and  x^  gives 

V,dx^-  Vodxoi-S  f^"^  Vdx  +X(  V^'dx^^  Vo'dx^)+8  r^XV'dx=0. 

71.  Cor.  It  may  be  shown  in  nearly  the  same  manner,  that  when 
fVdx  =  a  maximum  or  minimum,  and  also 

fV'dx  =  c        and         fV'dx  =  c', 
the  problem  may  be  solved  as  a  case  of  absolute  maxima  and  minima 
by  replacing  F  by  V-]-XV'  -\-X'  V"  where  X  and  X'  are  arbitrary 
constants. 

Applications. 

72.  We  will  now  illustrate  the  principles  already  explained  by  a 
few  examples. 

1.  To  find  the  nature  of  the  line  (lying  entirely  in  one  plane) 
which  is  the  shortest  distance  between  two  given  points. 
Let  XqIJq  be  the  co-ordinates  of  the  point 


,B 

Ar 


|r„  1^' 


x 


A^  and  x-^ijy^  those  of  B.     The  general  value 
of  the  length  of  the  arc  of  a  plane  curve  AB 

is  /  ( 1  +  -T-f  )  dx  taken  between  the  proper  0 
limits.     Hence  in  the  present  case  we  shall  have 

U  =  r^  Vdx  =  f'^ll-hi^)  dx=  &  minimum. 
^Xq  J Xq   \         dx^J 

Here   V  =  \\  -^--^'A  —  •'^(t)'  ^"^  consequently  by  formula  (a), 
the  solution  of  the  equation  6=0  becomes 

y  =  c^  -f-  c'. 
and  the  shortest  path  from  ^  to  .5  is  a  straight  line. 


MAXIMA  AND   MINIxMA   OF   ONE   VARIABLE.  475 

The  equation 

'  a^—  aQ  =  0     or      V^dx^  —  VqcIxq  +  Pfyi  —  Pq^q  =  0 
disappears  in  this  case,  since 

dxQ  =  0,  dx^  —  0,  (J^i  =  0,     and     SyQ  =  0, 

the  limiting  values  of  both  x  and  y  being  fixed. 

To  determine  the  values  of  the  constants  c  and  c'  we  have  the  two 

equations 

2/i  =  cx^  +  c\         and         y^  =  cXq  +  c' ; 

thus  the  solution  of  the  problem  is  complete. 

2.  To  find  the  line  of  shortest  distance  between  two  given  curves. 

Let  the  equation  of  the  curve  AB  be  yQ  =  FqXq (1), 

and  that  of  the  curve  CD, 


As  in  example  1, 


(2). 


.  • .  y  =  cx-i-  c\ 

0  x~ 

and  the  shortest  distance  is  still  a 

straight  line. 

To  determine  the  values  of  the  constants  c  and  c',  and  the  limiting 
values  a*Q,  «/o,  x^,  y^,  we  proceed,  as  follows : 

From  (1)  and  (2)  we  get  the  following  conditions  connecting  dxQf 
6yQ',  dx^  and  Sy-^,  viz. : 

^^<^  "^  Lt^ J  *  '^'^"  ^^o^-^o,     and     8y^  +  [Jj  dx^  =  t^dx^, 

dF\x, 
.     dx-^ 

[!]„='''  """^   [1],=  ''' 

• .  5yQ  =  {t^—  c)  dxQ,         6y^  =  {t^  —  c)  dxy. 


in  which 
A-lso 


t,  =  -j^,      and      /,  = 
dxQ 


476  CALCULUS  OF  VARIATIONS. 

Substituting  these  values  in  the  equation  a^  —  aQ  =  0,  and  replacing 
Fj.  Fq,  Pj,  Fq  by  their  values,  we  get 

(1  +  c^)^dx^  -  (1  +  c')^dxo  +  c  (1  +  c2)~*  (^  -  c)dx^ 

-c{\  +c2)~*(/o-0«?-^o  =  0. 

Now,  placing  equal  to  zero  the  coefficient  of  dxQ  and  dx^,  the  only 
arbitrary  increments  renriainiiig  in  the  equation,  we  get 

(1  +  c^)^  -h  c  (1  +  c^f^iii  -c)  =  0,     and 

(l+c2)*+c(l+c^)"*(fo-c)=0; 
or,  1  +  c^i  =  0  .  .  •  (3),     and     l+cto=zO  --  -  (4). 

These  two  equations,  with  the  following 

suffice  to  determine  the  six  quantities,  c,  c',  a-Q,  ^qi  ^i5  ^i* 

The  equations  (3)  and  (4)   show  that  the  shortest  line  UJS'  cuts 

both  curves  at  right  angles. 

73.  In  the  preceding  example,  suppose  the  given  curves  to  become 

straight  lines  perpendicular  to  the  axis  of  x.     Then  dxQ  =  0,  and 

rfiPj  =  0,  since  the  extremities  of  the  shortest  line  will  necessarily 

have  invariable  abscissae. 

.  n  dvn  -.  d)/-,  1 

Also     Iq  =  -~  =z  Qo  ,     and     t^  —  -—-  =  ao;     .' .  c  —  —  =  0; 
'I'^Q  dx-t  tff 

and  as  c'  is  now  indeterminate,  the  required  line  of  shortest  distance 
may  pass  through  any  point  of  AB. 

This  is  an  example  of  Exception  2. 

3.    To   find   the  form   of   the   function    y,   which    shall   render 


a  maximum  or  minimum. 


"<>('*'&*'■ 


Here  we  have 


MAXIMA  AND   MINIMA  OF  ONE  VARIABLE.  477 

and  therefore,  by  formula  (6), 


^=''i^^- 


Making  c  =  Z*»,  and  solving  with  respect  to  dx,  we  get 


l"dv 
dxz=  ^ 


This  comes  mider  the  binomial  foriii,and  therefore  is  integrable  when 


1         .  11 


an  integer   or   zero ;    that  is,   when   n   has   one  of  the  following 
values,  viz.  : 

,111.  1        1  1  1    X 

^'  2'  3'  4'       • '      ^''  ""  '  "2'    ~  3'   "■  i' 

As  a  particular  case  of  this  problem,  suppose  n  =  —  -; 


or  dx 


y^_y2    ■       y//^-  2/^  V^  —  y^ 


478  CALCULUS  OF  VAEIATIONS. 

1                -^  2y  i 

.  • .  X  -\-  c  =  -I'  versin     ~ (ly  —  y^)  . 

If  the  limiting  values  of  x  and  y  be  given,  then 

(Ixq  =  0,     Sijq  =  0,     dxj^  =  0,     6y^  =  0, 
and  the  equation  Gj  —  a^  =  0  disappears. 

To  find  the  two  constants  c  and  I,  we  have  the  two  equations 

1/  •  "'2^0         /I -, 

^0  +  ^  =  2      ^®^^"^     "1 '^^^  ~"  ^«  ' 

>  a-i  +  c  =  -^.  versin    -y-y^yi-yi^ 

and  if  a^o  =  0,     and     yo  =  ^^     then     c  =  0,     and 

^7  .    ~^2y         /^ r-  .  .  .  .  n\ 

X  —  -I'  versin      -^  —  v/lt/  —  y^  V^r 

74.  The  equation  (1)  of  this  last  example  exhibits  the  solution  of 
the  celebrated  problem  of  the  Brachydochrone^  or  the  curve  of 
swiftest  descent. 

Thus,  let  A  and  B  be  two  points  in  the  same  ver-y^     g- 
tical  plane,  and  let  it  be  proposed  to  determine  the 
nature  of  the  curve  APB^  along  which  a  heavy  body 
will  descend  from  A  to  B  (under  the  influence  of  the 
force  of  gravity  alone)  in  the  shortest  possible  time. 

Denoting  by  t  the  time  occupied  in  passing  from  A  to  any  point  P 
in  the  unknown  path,  the  co-ordinates  of  which  point  are  x  and  y; 
by  s  the  variable  arc  AP^  and  by  g  the  velocity  acquired  by  a  heavy 
body  falling  vertically  during  a  unit  of  time ;  then  it  is  shown  by 
the  principles  of  Mechanics,  that  the  velocity  acquired  by  the  body 
in  descending  along  the  curve,  (when  it  has  reached  the  point  P,) 
will  be  expressed  by 

V^,  and  also  by   -^^^-^M-j 


MAXIMA  AND  MINIMA   OF  ONE   VARIABLE.  479 

•  *•  /  y      ( 1  +  ^)  dx  =  B,  minimum  between  the  limits 

X  =  x^z=:  0,     and     x  =z  x^  =  AF. 

The  equation  (1)  represents  a  cycloid,  the  axis  DC  =.1  being 
vertical,  and  the  extremity  of  the  base  coincident  with  A^  the  point 
of  departure. 

75.  4.  Through  two  given  points  A  and  B^  draw  a  curve,  of  given 
length,  so  that  the  area  included  between  the  chord  AB  and  the 
curve  APB  may  be  the  greatest  possible. 

This  is  a  problem  of  relative  maxima  and  minima,  since  the  curve 
is  to  be  selected  from  a  particular  class,  viz. :  those  which  have  a 
given  length  /,  or  which  fulfil  the  condition 

Also  JVdx  =z  I    ^  ydx  =  a  maximum. 

*/Xq 

Therefore  by  the  method  of  relative  maxima   y'')^ 


and  minima,  we  have  A  ^  d  b 

=  V^dx^  -  Torfi-o  4-  ^  r^  Vdx 

Xq 

+  X(  V^,Jx^  -  V^ .  dx,  +  Sf'''  Vdx] 

V  Xq 

Here  the  limiting  values  of  both  x  and  y  are  invariable,  giving 
dxQ  =  0,     Sy^  =  0,     dx^  =  0,     Sy^  =  0. 


480  CALCULUS  OF  VARIATIONS. 

Also        r+xr'=.  +  x(.+|:)*=/(„|) 

Hence  the  equation     a,  —  o„     disappears,  and  formula  (6)  gives 


V+XV'  =  c  + 


0-£) 


.•,.-c)(l.gf  =  X    and    %=j^-. 

,  (v  —  c)dy 

or  dx=     J— I—       ^%:^,  whence 

ic  =  —  [X2  -  (y  -  c)2f  +  c'     or     {x  ~  c^^  +  (3/  -  c)2  =  \^ 
and  the  required  curve  is  the  arc  of  a  circle. 

To  determine  the  constants  c,c\  and  X  we  have  the  three  equations 
(^0  -  c'f  +  {Vo  -  cf  =  X2,     {x,  -  cy  +  (yi  -  cy  =  X2     and 

i  chord  AB  -l 


=  sin(^), 


or  when  the  origin  is  at  A  and  the  chord  AB  coincides  with  the 
axis  of  ic, 

c'2  4-  c2  =  X2,  (x^  —  c')2  -f-  c2  =  X2,    and    -i  =  sin--. 

^X  «iX 

76.  5.  Given  the  length  /  of  the  curve  joining  two  fixed  points 
A  and  B,  to  find  the  form  of  the  curve  when  the  surface  generated  by 
its  revolution  about  the  axis  AB  is  the  greatest  possible. 


MAXIMA  AND  MINIMA  OF  ONE  VARIABLE. 


481 


nerefVci.=ly,[l+'£:j,.     a 


=  a  maximum, 

«.d/F'<^=/;^'(.+|-:)*..=/. 

.-.  J)U=zI>f{V  +  XV')dx  =  0  and 

v+xP...,(,.|:)Vx(..g)*=4.|). 

The  equation  a^  —  aQ  =  0  disappears,  and  (b)  gives 

•••(■H-f.)'=-'+^.|:=*^"-^-- 

To  integrate  this  put  2'ry  +  X  =  2     and     y'^FZT^  =  2;  —  ^, 


and 


'/(2*y  -f-  X)2  -  c2  = 


;2- ^2 
2^ 


^^  =  -2^'T 


'2'n-     ^  i        2'Ji'     '^2'rfy  -f-  X  —  y/(2^_j.  x)2  —  c« 
e  -      27r'y  -4-  X  -f  v^-n-y  +  X)2  -  c^ 


2'K 


log 


=  C71og 


y+CJ'  "t  VW^  c'f  -  C'» 


C" 


31 


482  CALCULUS  OF  VARIATIONS. 

in  which    C  ::=  ^,  (7'  =  ^,     and     C"  =  ~ 

This  is  the  equation  of  the  Catenary,  which  therefore  is  the  required 
curve. 

77.  Prop.  To  find  the  form  of  the  function  and  the  values  of  the 
limits  Xq  and  x-^  which  shall  render 

U  ~  V  -\-  I    ^  Vdx  a  maximum  or  minimum,  where 
*JXq 

The  general  equation  6  =  0,  being  derived  exclusively  from  the 
terms  under  the  sign  of  integration,  must  be  the  same  as  in  the  last 
proposition,  and  therefore  it  will  be  necessary  to  consider  only  those 
terms  which  refer  to  the  limits : 

Fut  0?  F  =  M'dx,  +  N'dy,  +  P,'d  i^\  +  A'<^(0)  +  &c. .  •  . 

Then  the  additional  terms  in  D  U,  resulting  from  F',  are 

and  the  first  member  of  the  equation  aj  —  Oq  =  0  will  be  increased 
by  these  terms,  which,  being  of  the  same  form  with  the  terms  pre- 


MAXIMA  AND   MINIMA  OF  ONE    VARIABLE.  483 

viously  found  in  that  equation,  there  will  be  no  difference  in  the 
manner  of  discussing  it  in  its  modified  form. 

It  must  be  remembered,  however,  that  the  possibility  of  satisfying 
the  condition  1)17  =  0,  depends  upon  the  fact  that  the  number  of 
independent  increments  in  the  equation  a^  —  ao  =  0,  does  not  usually 
exceed  the  number  of  arbitrary  constants  in  the  integral  of  the 
equation  b  z=  0.  Hence  if,  in  any  particular  case,  the  number  of 
independent  increments  should  be  greater  than  the  number  of 
constants,  the  solution  would  be  impossible. 

Now  in  the  case  at  present  under  consideration,  the  number  of 
increments. 

"-*-(§). m. 

relating  to  the  inferior  limit  is  n'  -f-  2 ;  and  the  number  of  incre- 
ments already  found  to  exist  in  ag  i^  n  -f  1. 

If  then  7i'-|-2>^+l}  or  n'  ^  n  —  1,  the  solution  of  the  prob 
lem  will  be  impossible. 

Similar  remarks  apply  to  the  superior  limit ;  and  we  conclude 
that  when  the  new  function  V  contains  any  coefficient  of  an  order 
higher  than  n  —  1,  the  function  U  will  not  admit  of  a  maximum  or 
minimum. 

78.  Prop,  To  find  the  form  of  the  function  y  and  the  values  of 

the  limits  Xq  and  x^,  which  shall  render  ^  —  J    ^  Vd^  a   maximum 

or  minimum,  where 

^=^  t  ^'  I  •  •  •  •  S '-  y-  (Do-  •  •  •  (S)o'  ^-  y- 

\dx)^ [(ix'^'U 

The  general  equation  DU  =  0  becomes  in  this  case    (p.  411) * 


'484  CALCULUS   OF   VARIATIONS. 

+^.''"'-](t)/^-- 

This  being  written  in  the  form 

«i  —  «o  +  /    ^  h^ydx  =  0, 

shows  that  b  is  the  same  as  before,  and  therefore  the  form  of  the 
function  y  is  not  changed  by  supposing  Fto  contain  explicitly  the 

limiting  values  of  a:,  y^  -f-,  &;c. 

Also  the  terms  in  a^  —  CTq  =  0  are  of  the  same  nature  as  if  V 
did  not  contain  the  limits,  forming  a  series 

A^dx^  4-  ^i%i  +  C^  ('^^j  +  &c.  4-  A^dx^  +  ^0(^1/0  +  Co  (^)  &c. 

i4|,  ^1,  Cj,  &;c.,  Jq,  ^0?  ^05  ^^'1  teing  constants.     For  in  the  ex- 
pressions 

/    ^  m-^dx^    I    ^  Mffix,  &c., 

*f  Xq  ^  Xq 

the  same  supposition  is  made  as  in  the  terms 


M.AXIMA    AND   MINIMA   OF   ONE   VARIABLE. 


485 


and  the  other  coefficients  of  the  several  increments  in  the  equation 
a^  —  Oq  =  0,  where  V  did  not  contain  the  limits ;  viz.  :  that  the 
value  of  y,  derived  from  the  equation  b  =  0,  has  been  substituted  in 
m,,  mo,  &;c.    This  substitution  being  effected,  and  the  definite  integrals 


being  formed,  the  quantities  A^^  B-^,  Aq^  B^^  &c.,  will  become  entirely 
constant. 

Thus  the  mode  of  treating  the  equation  i>  t^"  =  0  is  in  all  respects 
the  same  as  in  the  case  previously  considered. 

The  following  examples  will  illustrate  the  cases  considered  in  the 
last  two  propositions. 

79.  Ex.  Having  given  the  area  c  of  the  figure  BAA-^B^^  bounded 

by  the  axis  of  ar,  by  two  ordinates  passing  through  the  given  points 

B  and  J5,,  and  by  a  curve  ACA-^^to  find  the  nature  of  the  curve  and 

the  values  of  the  extreme  ordinates  BA  and  BiA^,  when  the  peri. 

meter  of  the  figure  is  a  minimum.     Put 

OB=x,,    OB,=x,,  BA=y,,   B,A,=y^. 

Then,  sin(ie 

BB^  =  x^  —  Xq 

is  constant,  we  have 


BA  +  B,A,  +  ACA^  =  yo  -f  yi 
Also  P'  V'dx  =  f%dx  =  c. 

*^  Xq  V  Xq 

.-.    U=  V"-\-  r\V+\V')dxz=amim 

V  Xn 


mmimum. 


mmimum. 


Here  U  contains  a  term  V"^  exterior  to  the  sign  of  integration, 
involving  the  limiting  values  of  y,  and,  therefore,  by  the  method 


486  CALCULUS  OF  VARIATIONS. 

applicable  to  such  cases,  combined  with  that  of  relative  ma.cima 
and  minima,  we  have 

Now     V  -{-  "kV  =  fly,  —  j,  and  therefore  by  formula  (6) 


dx 

Put    ^^  =  /3,     and     i^a,      then      ^1  +  g)  (,/3  -  y)^  =  «2 ; 

{^—y)dy  1 

.•,dx  =  ^^^^^— --^^-— —     and    a:  =  ^2  +  [oc2  -  (^  _  yf^ 

or,  (a:  —  Cg)^  +  (y  —  i^)^  =  '^^j  the  equation  of  a  circle. 

Hence,  the  curve  ACA^,  is  a  circular  arc. 

To  determine  the  values  of  the  ordinates  yg  ^^^  Vii  ^^^  ^^^^  ^^ 
a,  the  radius  of  the  circle,  we  recur  to  the  equation 

ttj  —  ccq  =  0,  which  becomes,  in  the  present  case 

+  iV^'%i+iV^%o  =  0,     (1), 

since  V+W  does  not  contain  Pg?  -^3?  ^^-f  ^"d  ^"  contains  only  y^ 
and  yp 

Also,  since  the  points  B  and  B^  are  given,  G?a:,  =  0,  and  dx^  =  0. 
Thus,  (1)  is  equivalent  to  the  two  conditions 


MAXIMA  AND   MINIMA  OF  ONE  VAKIABLE. 


487 


But 


dV"  dV" 

N'=^=\,     and     N"=^=l. 

dyQ  dy^ 


Hence,  by  substituting  the  values  of  iV^',  N"  and  P^,  we  obtain 

■■■(1).=  +-  ■■'  0)  =  -- 

And  therefore  the  arc  ACA^  is  a  semicircle,  the  tangents  at  A  and 
A^  being  perpendicular  to  OX. 


Also, 


radius  a  = -(iCj—aro),     and     yi=  y^. 


But  area  BAA^By^=z  2(1- yQ+  -'ira?  =  c,  and  .  ' .  yo  becomes  known, 

thus  making  the  solution  complete. 

80.  £x.  To  find  the  curve  of  swiftest  descent  from  one  given  curve 
to  another,  the  motion  being  supposed  to  commence  at  the  upper 
curve. 

Let  AB  and  A^B^  be  the  given  curves,  and 
CCi  the  curve  required. 

Put     OD  =Xq,     I)C=  yo,     OE  =  x, 
EP  =  y,     OF=x,,     FC,=  y,,     CP=s. 
Then,  by  the  principles  of  Mechanics  (before 
cited),  the  velocity  acquired  by  the  body  in 
descending  from   C  to  P  along  the  curve   CPC^,  is  expressed  by 

^. 


/^ F7T        /^ — ; ;  ^    .      .     ds       dx     f       du^ 

^2yy  IP  =  /2y(y^yo)',     and  also  by  —  =  —  W  1  + 


dt       dt 


rf«  =  [%(3,-y„)r^.[l  +  £-]V 


minimum. 


488  CALCULUS  OF  VARIATIONS. 

Here  F  contains  the  limit  yg  explicitly;  and  therefore  DC'' will  c^»d 
tain  the  additional  terms 

[X? "« (SW  ^"'^  t-^? "» ''']  ^^° 

which  terms  appear  in  the  equation  a^—  aQ  =  0,  but  not  in  the  equa- 
tion 6  ==  0. 
Also,  since  F  =  /(y,  -f- j,       we  have,  by  formula  (6), 

_.  rfy_^  rae_(y-y„)-i^ 

■   ■  u!j:       L      y  —  yo      -I 
This  is  the  differential  equation  of  a  cycloid  having  the  axis  parallel 
to  y,  the  cusp  or  extremity'  of  the  base  at  the  upper  point  «»,  y„  and 
the  diameter  of  the  generating  circle  =  2C. 
The  equation  Oj  —  o,  =  0  gives,  in  this  case, 

+  (7^^'no^^)^yo  =  0....(i). 

But    no  =  3—= T-=  — iV^= 1-^,    smce    JV^ t^  =  0 

ayo  «y  <^«  or 

.',fn,dx=-  f-^  dx^-P.i-c,, 


MAXIMA  AND   MINIMA   OF   ONE   VARIABLE.  489 

r.f^^n,dx-^{P,\-{l\\,     and     fi^f^ri.dx 

Again,  if  the  differential  equations  of  the  two  given  curves  be 

1:='°'  ^""^  lf='" 

we  shall  have  the  following  conditions  connecting  the  values  of  dx^, 
6^Q,  dx^,  and  dy^,  viz. : 

^Vo  +  (^ )  ^-^0  =  ^0^-^05     and     5y^  '^\£\  ^^^—  ^i^-^i* 

Now  substituting  the  values  of  ^y^^  Sy,,  I    ^n^dx^  and    /    M-^l  iirdx 

*J  Xq  J  Xq   \dx/Q 

in  (1),  and  placing  the  coefficients  of  dx^  and  dx^,  separately,  equal  to 
zero,  we  get 

r.  +  (PA[^.-g)J  =  0,    and 

-[(A)o-(A),]['»-ffi)J=<''    -> 

-    [(-g-)*-<--)-*]r©I(-£)^- 


490  CALCULUS  OF  VARIATIONS. 

From  (2)  we  obtain        l  +  ^i(;r^)  — ^j     ^"^  therefore  the  cycloid 

intersects  the  second  curve  at  right  angles. 

Also,  from  (3)  we  get         1  +  ^o  ij)  =0;         .'.  t^  =  to, 

and  the  tangents  to  the  two  curves,  at  the  points  of  intersection  with 
the  cycloid,  are  parallel.  The  co-ordinates  of  those  points  are 
readily  found. 

81.  Prop.  To  determine  the  forms  of  the  functions  y  and  z,  and 
the  values  of  the  limits  x^  and  Xq,  which  shall  render 

U  =^  I    ^  Vdx  a  maximum  or  minimum,  where 
Jxq 

r         dy      d'^y  d'^y  dz       d'^z  d^z'X 

The  equation  1)17=0  becomes  in  this  case 
V,dx,  -  V,dx,  +  \_P,  -  ^'  +  &c.]  Sy,  -  [p,-^2^  &c.]  .  oy, 

+[^--].(a-[A-.o,,(f)^....4n53]^ 
+(-')-'^]^^''- 

d"'P  "H  « 


MAXIMA  AND  MINIMA  OF  ONE  VARIABLE.  491 

If  the  functions  y  and  z  be  independent  of  each  other,  their  varia- 
tions hy  and  ^z  will  also  be  independent ;  and,  by  reasoning  as  in 
previous  propositions,  it  will  appear  that  we  shall  have  the  conditions 


dx         dx^  ^        '       dx'^  ' 

dP  '       d'^P'  d^P   ' 

iVr'_^  +  lil2_&c....+(-l)-.^  =  0...(l). 

dx  dx^  ^\         )         ^^m  \   ) 

And  for  the  equation  of  the  limits 

V^dx^-  r„<&„+  [a-  g  +  &c.]  ^y^  -  [p,  -  ^  +  &c.]^iyo 

+  [A'  -  &c.]  ("f f)^  -  IP'  -  &c,]„  ( J)^-  &c.  &c.  =  0 (2;. 

The  mode  of  treating  these  equations  is  exactly  the  same  as  that 
employed  when  V  contained  but  one  function,  and  by  reasoning,  as 
in  that  case,  it  may  be  readily  shown  that  the  number  of  equations 
applicjible  to  the  solution  of  the  problem  will  not,  in  general,  be 
affected  by  any  equations  of  condition  restricting  the  limits.  For 
every  such  equation  of  condition  will  diminish  by  unity  the  number 
of  terms  in  (2),  either  by  reducing  to  zero  the  variation  which 
appears  in  such  term  ;  or,  by  uniting  two  terms  in  one,  and  thereby 
diminishing  by  unity  the  number  of  equations  deducible  from  (2). 

But  the  given  equation  of  condition  will  just  supply  the  place  of 
that  which  has  disappeared. 

Thus  it  will  suffice  to  prove  that  (1)  and  (2)  furnish  the  requisite 
number  of  equations  in  a  single  case,  as  when  the  limits  of  x  are 
alone  fixed. 


492  CALCULUS  OF  VARIATIONS. 

Now  the  first  of  equations  (1 )  is  of  the  order  2n  in  y,  and  m  -\-  n 
in  0,  and  the  second  of  equations  (1)  is  of  the  order  m  H-  n  in  y, 
and  2wi  in  z.     They  are  therefore  of  the  forms 

^r         dy  d'^'-^^y       dz  d^"'z'] 

If,  then,  we  differentiate  (3)  2m  times,  and  (4)  m  +  n  times,  we 
shall  have  3///-  -f  ?*  +  2  equations  with  which  to  eliminate  the  3wi-j-n 

quantities     ^, -,— sm+n'     ^  ®  resulting  equation  will  be 

of  the  order  2m  -{-  2n  in  y.  The  integral  of  this  equation  will  con- 
tain 2m  +  2u  constants.  But  the  number  of  equations  given  by 
(2)  is  exactly  2/i  +  2?>?,  viz.  :  the  2ri  equations, 

and  the  2m  equations, 

[P2'-&C.]i   =r  0,  &C. 

Plence  the  problem  is  in  general  determinate,  but  there  are 
exceptions  entirely  similar  to  those  considered  in  the  case  of  a  single 
dependent  function  y. 

82.  If  the  functions  y  and  z  be  connected  by  an  equation  L  =  0, 
and  if  it  be  possible  to  resolve  that  equation  with  respect  to  y  cr  s, 

so  as  to  obtain  a  result  of  the  form  z  =/l.r, y,  j-,  dec.  j,    the  values 

dz      d.^z 
of  — ,     —-,  &c.,  can  be  formed  by  differentiation,  and  substituted  in 

that  of  F,  which  will  then  contain  ar,  y,  and  the  differential  coeffi- 
cients of  y  with  respect  to  ar,  thus  presenting  a  case  already 
considered. 


MAXIMA  AND  MINIMA  OF  ONE  VARIABLE.  493 

83.  But  since  the  proposed  equation  L  z=  0  is  often  a  differential 
equation  difficult  to  be  integrated,  we  are  often  compelled  to  adopt 
the  method  already  noticed,  (Page  444)  in  which  by  the  introduction 
of  a  new  indeterminate  quantity  X,  and  a  suitable  determination  of 
its  value,  we  are  enabled  to  obtain  an  expression  for  (^tT"  which  shall 
contain  but  one  of  the  variations  §7/  and  6z  under  the  sign  of  inte- 
gration. 

Thus,  if  we  denote  by  ^,  the  sum  of  the  terms  exterior  to  the  sign 
of  integration  in  the  value  ofSV,  (Page  445)  there  will  result 

and  if  we  so  assume  the  quantity  X  as  to  fulfil  the  condition 

dx 
it  will  appear  by  reasoning,  similar  to  that  employed  when  y  was 
the  only  function,  that  the  condition  6U  =z  0  cannot  be  satisfied  (so 
long  as  the  form  of  6y  is  arbitrary)  unless  we  have  the  two  conditions 

^=0     and     iV+Xa-^t^-^4-&c.  =  0. 

dx 

Hence,  we  have  for  the  solution  of  the  problem,  the  three  general 
equations 

and  i,'  +  x.'-ll^±^  +  &c.  =  0. 

ax 

which  are  just  sufficient  to  determine  the  three  unknown  quantities, 
X,  y  and  z. 

84.  We  will  now  give,  in  conclusion,  examples  to  illustrate  the 
cases  and  methods  above  explained. 


494  CALCULUS  OF  VARIATIONS. 

Ex,  To  find  the  nature  of  the  line  which  is  the  shortest  distance 
between  two  given  points  in  space  there  being  no  restriction  by 
which  the  line  is  required  to  be  confined  to  one  plane. 

The  general  value  of  the  length  of  the  arc  of  a  curve  of  double 
curvature  is 


/(-£-a- 

iken 

.  between 

I  the  proper  limits. 

Hence  in  the  present  case  we  shall  have 

U 

=y;:'('+s+ 

-—  1  dx  =a  minimum. 
dx^l 

Here    V^ 

('-2-3' 

dt, 

0,.'  =  -. 

:0 

P, 

dV 
dx 

dy 

dx 

p,      dV 

dz 
dx 

"*  I 

1         dy"-       dz^' 

vA  +  f- 

~d^ 

dx^ 

Pg  =  0,     P^'  =  0,  &c. 
Hence  the  equations 

N 
become 

or    Pj  = 


--^H-  (fee. 
dx 

=  0 

and 

N'  - 

dP,' 
dz 

+  &C.  = 

=  0 

dp, 

dx 

=  0 

and 

dP,' 
dx 

=  0 

dy 

dz 

dx 

c    and     P; 

dx 

y'+f- 

dz"^ 

C^2 

/ 

V 

dx^ 

d^ 
^  dx'^ 

=:  C 


Eliminating  first— and  then -^  between  these  two  equations,   w« 
readily  obtain  results  of  the  forms 


MAXIMA  AND  MINIMA  OF  ONE  VARIABLE.  496 

dy  dz 

•—=in    and      —•  =  n   in  which  m  and  n  are  constants, 
ax  ax 

,' .  y  z=z  mx  4" Pi         and         z  =l  nx  •\-  q. 

These  are  the  equations  of  a  straight  line,  which  therefor.e  is  the 
vnortest  distance  required. 

To  find  the  values  of  the  constants  m,  w,  jr;,  and  q^  we  introduce 
the  given  limits  x^^  y^?  ^g?  ^\i  Vi^  ^u  ^^^^  thus  get 

which  suffice  to  determine  7n,  7i,^  and  q. 

85.  If  the  limiting  values  of  x  only  were  given,  those  of  y  and  z 
remaining  indeterminate,  the  terms  exterior  to  the  sign  of  integra- 
tion would  give 

(P.)i  =  0>  (^1)0  =  0,  {P,'\  =  0,  (P/)„  =  0, 

which  are  equivalent  to  the  two  equations 

m  =  0     and     w  =  0, 

thus  leaving  the  other  two  constants  p  and  q  indeterminate,  and  pre- 
senting one  of  the  cases  of  exception  already  noticed. 

86.  JEx,  To  find  the  shortest  distance  between  two  given 
surfaces. 

Let  the  equation  of  the  first  surface  be  /o(^o?  ^o^  ^n)  ==  0  •  •  •  •  (1) 
and  that  of  the  second  surface  /i(^i?  Vii  ^1)  =  ^  "  • '  (2) 

and  we  immediately  deduce  as  before 

y  =  mx  -\-  p (3),  z  =  nx  -{•  q (4) 

which  show  that  the  shortest  path  is  still  a  straight  line. 

To  fix  the  co-ordinates  of  the  extremities  of  this  line  we  form  the 
complete  increment  of  (1)  and  (2)  thus  : 


496  CALCULUS  OF  VARIATIONS. 

LdxQ       d;/^  \dj/Q       dz^  \dx)j      ^       dy^ '      ^      dz^       ^  ~ 

\dx^       dy^   \dxj^       dz^    \dx)^\  dy^     ^^      dz^      ^  ^  ^ 

Put  for  brevity 

^  ^  ^/q  jfi 

dyQ  dy.  dz^  dz. 

dxQ  dx-^  dxQ  dx^ 

their  values  derived  fforn  equations  (3)  and   (4).     We  shall  thus 
ibtain 

(1  +  mmQ  +  wwo)  dxQ  +  Wo^yo  +  ^'o^^o  =  ^ 

(14-  wzwij  +  nn^)  dx-^  +  ^i*^//!  +  n-^^z-^  =:  0. 

Now  eliminating,  by  the  aid  of  these  equations,  dx^  and  dx-^,  from 
the  equations 

F„rfx„  +  (P,)/y„  +  (P/)„fe„  =  0 

and  placing  equal  to  zero  the  coefficients  of  ^y(„  8zq^  Sy^^  §z^,  we 
obtain 

ttIqVq  -  (Pi)o  (1  +  mirif^  4-  >iWo)  =  0  • (7) 

^i  ^1  -  (^i)i  (1  +  mm,  4-  w/?i)  =  0 (8) 

Wo  1^0  -  (A')o  (1  +  mm,  +  n^o)  =0 (9) 


f^i  -  (A')i  (1  +  wiT^i  +  n«i)  =  0 (10). 


If  now  we  replace  Vq  and  (Pi)o  &c.  in  (7),  (8),  (9)  and  (10),  by 
their  values 

(1  -f  m2  +  w2)*     — &c. 

y'l  4"  m^  -f-  '/t^ 


MAXIMA  AND  MINIMA  OF  ONE  VARIABLE.  497 

we  readily  find  from  (7)  and  (9)      m  =  mg,  n  =  Wo (11) 

and  from  (8)  and  (10),     m  =zm^     and    n  =  w^ (12). 

Now  eliminating  Xq^  i/q,  Zq^x^^  y^,  s^,  which  quantities  occur  in  the 
values  of  Wq,  Wq,  mj,  and  n^,  by  means  of  the  six  equations, 

Vo  =  m^Q-rP,    Vi  =  ^a^i  +  P, 
Zq  —  uXq  -^  q,     z^  —  7ia?i  +  q, 

there  will  remain  the  four  equations  (11)  and  (12)  with  which  to 
Compute  the  values  of  tw,  n^  /?,  and  q ;  thus  the  line  of  shortest 
distance  will  be  fixed  in  position ;  and,  by  combining  its  equations 
with  those  of  the  given  surfaces,  we  can  find  the  values  of 

87.  The  equations  (11)  and  (12)  show  that  the  line  of  shortest 
distance  is  normal  to  both  surfaces.  For  the  assumed  values  of 
m^  and  %  indicate  that  they  represent  the  tangents  of  the  angles 
formed  by  the  projections  of  the  normal  to  the  first  surface  on  the 
planes  of  xy  and  xz  with  the  axis  of  x ;  while  m  and  n  denote  the 
tangents  of  the  corresponding  angles  formed  by' the  projections  of 
the  line  of  shortest  distance. 

A  similar  remark  applies  to  the  quantities  Wj  and  Wj,  and  the 
normal  to  the  second  surface. 

88.  JEx,  To  find  the  shortest  distance  traced  on  the  surface  of  a 
given  sf)here  betw;een  two  given  points  in  tlie  surface. 

Here  the  quantity  to  be  rendered  a  minimum  is  the  same  as  in 
the  last  tw  :>  examples,  viz. : 

but  since  the  path  is  restricted  to  the  surface  of  a  given  sphere,  the 

32 


498  CALCULUS  OF  VARIATIONS. 

co-ordinates  a:,  y,  and  2;,  of  any  point  in  the  required  path,  will  be 
connected  by  the  relation 

z^  +  y^ -\-z^  =  r^,     or     L  =  x  +  y^-{- z^  =  0 (2). 

Hence  the  variations  of  y  and  z  will  not  be  independent  of  each 

other. 

dz 
Now  we  might  form  from  (2)  the  value  of  — ,  which,  substituted 

in  (1),  would  reduce  V  to  a  form  in  which  it  would  no  longer  con- 
tain the  function  z,  or  its  differential  coefficient,  or  we  may  adopt  the 
method  of  Lagrange,  which  is  usually  the  easier.  Taking  the  second 
method,  we  have 


■-=(■+£+£)*• 


dy  dz 

dx  dx 

•n T>  t 


d^ 

dx^  '   dx^ 


V^  +  ^  +  rfJ^         V^  +  rf^^-^ 

,^      dV      ^      ,^,      dV      ^ 
dy  dz 

dL       dy      rt         d^  ,      d^       n, 

dy       dx'     ^         dy^       ^'  dx'     ^ 

dx 

Hence  the  equations    iV-fXa ^— i^ ^ -f  &c.  =  0, 

ud  jr+x«'-  '^^^''^'^^'^  +  &c.  =  0, 

oeoome,  in  this  case, 

dx       dx  dx         dx 

^  dz       dP'      ^dz         dX      ^ 
dz        dx  dx        dx        ' 


dK   .    d   \  dx 


MAXIMA  AND  MINIMA  OF  ONE  VARIABLE.  499 

=  0....(3), 

=  0....(4). 


ys+ 

■TA 

^y/^*1> 

dz' 
dx' 

z^  + 

d 

dx 

1 

dz 
dx 

-  dx^ 

W'-'£* 

dz'' 
dz^ 

dk 
Eliminating  -7-  between  (3)  and  (4),  we  get 


^  \                /               ^ 

dx  1^1                 ^^ 

/      d^  dz^  I       ^  <^^  \  f7~~df-      ^ 

and  by  integration 


d    \  dx  I  d    \  dx  |_A. 

dx  I       I      Ihh       1^  I  dx  '         ~~       ~       ~     ' 


^ 


dy         dz 

=  c....(5); 


dx         dx 


1   4.  'J^  -L  ^ 

'^  dx'''^  dx' 


or,  by  changing  the  independent  variable  from  x  to  «,  (5)  becomes 

dy  dz  ,„. 

z-j-  —  y-r-  =c (6). 

ds      ^  ds  ^  ^ 

By  similar  reasoning  we  may  obtain 

dx         dy  ,^.  .        dz         dx 

Multiplying  (6)  by  ar,  (7)  by  «,  and  (8)  by  y,  and  adding,  we  get 

ex  4-  c^z  +  Cjy  =  0,    or    z -{ x  -f-  —y=  0  .  .  .  .  (9), 

the  equation  of  a  plane  passing  through  the  origin. 

Thus  the  required  line  of  shortest  distance  on  the  surface  of  the 
sphere,  is  confined  to  a  plane  passing  through  the  centre,  and  is,  con. 
sequently,  a  great  circle. 


600  -         q4:LGULUS  OF  VARIATIONS. 

The  equation  a^  -   ao  =  0  in  this  case  disappears,  since 
dxQ  =  0,     dx^  =  0,     dj/Q  =  0,     ^y^  =  0,     6zq  =  0,     and     Sz^  =  0. 

The  constants  —  and  — ^  are  found  by  substituting 

^o,yo.^o,     ani     arj,  y^,  2?i,     for     a?,  y,  and  e  in  (9) . 

89.  If  the  limiting  values  of  x  only  were  given,  or  the  problem 
that  in  which  it  is  required  to  find  on  the  surface  of  the  sphere,  the 
shortest  path  between  two  parallel  sections,  the  variations  6i/q,  dy^, 
(52o,  (J^i,  would  not  reduce  to  zero,  and  the  equation  Oj  —  a^  =  0 
would  give  the  four  conditions 

(P,+  >.^)„=0,  (P,+  X^)i=0,  (/>,'+ X/3')o=0,  (A'+X^')i=0. 

dy 

or,  I   ^=^=:==.I  +  X„y„  =  0--..(10); 


/!  +  -  +  - 


dx'^      dx^ 


dz 


and  I   —- ^— ^-  I  +  X„.„  =  0.  . . .  (11)  ; 

,  V  dx^       dx^ 


0 


which  apply  to  the  inferior  limit,  with  two  similar  equations  for  the 
gruperior  limit. 

Eliminating  \q  between  (10)  and  (11),  there  results 


,\/ 


dy  dz 

=  0, 


dx  dx 


dx"^       dx^l  0 


Hence,  the  constant  c  =  0  in  (5) ;  and  that  equation  becomes 

dy         dz       .  dy       dz 

.   .  2?  /.  —  y  —  =  0 ;     or,      -^  =  ^. 
dz         dx  y        « 


(J 


MAXIMA  AND   MINIMa   OF   ONE   VARIABLE. 


501 


.  • .  log  y  ■=.  log  z  +  log  m  =  log  mz ;     and     y  =  mz. 

This  is  the  equation  of  a  plane  passing  through  the  axis  of  x,  and 
forcoing  an  arbitrary  angle  (tan-^m)  with  the  plane  of  xz-  Hence, 
the  required  path  is  the  arc  of  any  great  circle  perpendicular  to  the 
planes  of  the  parallel  sections. 


aT-rc 


^^-i. 


'Sj. 


^x-^ 


M  . 


vW 


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OCT  29  1942 


APR    28  1948 


,..-  o.A^ 


aAii^'SOAT' 


SEP   13    1944 


AUG    1719* 


TOlanSTW 


^^^    80I84S 


PEC    a     1945 


-—i^^ 


^<i 


w 


onir98i9. 


RECDUJ 


mui 


19S9 


MAR  19  1347 


APti  14  mi 


'^OV   la  tfl47 


LD  21-100m-7,'40(6936s)l 


-Wv^V 


